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THE 
MYSTERY OF SPACE 



The domain of the senses, in nature, is almost 
infinitely small in comparison with the vast 
region accessible to thought which lies beyond 
them. — Tyndall, "On Radiant Heat." 



THE 

MYSTERY OF SPACE 

A Study of the Hyperspace Movement 

in the Light of the Evolution of 

New Psychic Faculties 

and 

An Inquiry into the Genesis and 
Essential Nature of Space 



BY 

ROBERT T. BROWNE 




NEW YORK 
E, P. DUTTON & COMPANY 

68 1 Fifth Avenue 



Copyright 19 19, by 
E. P. DUTTON & COMPANY 



All Rights Reserved /^) Q Lqi 



k : ti 1919 



Printed in the United States of America 



©CI. A 5 3 6 74 8 



TO 

THE CHERISHED MEMORY 
OF 



it P* |fr* 



WHOSE WIFELY DEVOTION, 

SYMPATHETIC ASSISTANCE AND ENCOURAGEMENT 

DURING THE EARLY LABORS ON THE TEXT 

WERE A CONSTANT SOURCE OF 

INSPIRATION AND FORTITUDE TO THE AUTHOR, 

THIS BOOK IS DEDICATED 

AFFECTIONATELY 



PREFACE 



Mathematics is the biometer of intellectual evo- 
lution. Hence, the determination of the status quo 
of the intellect at any time can be accomplished most 
satisfactorily by applying to it the rigorous measure 
of the mathematical method. The intellect has but 
one true divining rod and that is mathematics. By 
day and by night it points the way unerringly, so long 
as it leads through materiality; but, falteringly, blindly, 
fatally, when that way veers into the territory of 
vitality and spirituality. 

Wherefore, when we have wished to ascertain the 
real status of the intellect, as well as its limitations, 
tendencies, possibilities, we have turned to its conduct 
in the field of mathesis where it is least trammeled in 
its ingressive and egressive motivations because of the 
natural and easy accommodation which is offered for 
intellectual movement. Whether there are signs of 
moribundity or symptomatic evidences of marked 
growth or of a termination of intellectual regnancy, 
or whatever may be the occasion for the examination, 
no surer index than the mathematical may be found 
for the purpose. Full logical justification is, therefore, 
claimed for the choice of mathematical evidences to 
test the assumption that a new era of conscious mental 

vii 



viii PREFACE 

conquest is opening for the vanguard of humanity and 
sequently for the human family. 

The treatment of these evidences has fallen log- 
ically into two divisions, namely, the first, a brief 
and elementary review of the principles of the non- 
Euclidean geometry and their bearings upon the ques- 
tion of space as the subject of mathematical study; 
and second, the consideration of space as a psycho- 
logical, vital and dynamic or creative phenomenon. 

In Part I an effort has been made to trace the 
growth of the notion of hyperspace and to show that 
it is a symbol of a new epoch of intellectual expansion, 
an actual seizure of a new domain of awareness by 
the mind. And for this purpose a critical examina- 
tion of the fundamental question of dimensionality is 
entered upon from which it develops that the status 
of this primary norm of mathematical thought exhibits 
a relatively inchoate character because of its insuffi- 
ciency as a definitive quantity, and further, because of 
its rather superfoetated aspect when utilized as a 
panacea for mathetic symptoms. Also, it has been 
found necessary to survey the field of the four-space 
which has been accorded such a prominent place in 
the mathematical thought of the day. The reader 
should find in the chapter devoted to it adequate mate- 
rial for thought and sufficient comprehension of its 
meanings as a mathematical contrivance. 

In Part II an endeavor is made to interpret the 
evidences offered by high mathematic considerations 
in the light of the larger psychogenetic movement. 
For this undertaking the quality of awareness has been 
studied with the view to establishing its priority as a 
determining agency in the consideration of space in 



PREFACE ix 

aspects broader and less restricted than those em- 
braced in the mathematical premises. Wherefore, it 
appears that there are massive implications arising 
out of the hyperspace propaganda which have hitherto 
been neglected in the preliminaries incident to the 
fabrication of its structure. A very brief, and more 
or less symbolic, sketch of the genesis of space has 
served to demonstrate its essential nature as a con- 
substantive of materiality, vitality and intellectuality, 
the three major objective processes. Consequently it 
becomes imperatively necessary that any view of space 
which neglects its kosmic intent and purpose should 
be regarded as gravely fragmentary and insufficient. 
It is only by linking up the two aspects of space, the 
mathematical and the psychological, in such a manner 
that the one shall supplement the other, that we shall 
be able to arrive at a truly satisfactory understanding 
of its nature. 

In Chapter IX attention is invited to some of 
the extremities of mathematical laws wherein it is 
shown that, because mathematical goods are strictly 
of intellectual texture and fabric, vain is the hope of 
reaching any reliable certainty with respect to many 
vital questions, even regarding space itself, by means 
of 'the mathematical method. The intellect, and, 
therefore, mathematics encounter the most formidable 
stricture when effort is made to maneuver in the field 
of vitality or realism. In addition, it is shown that, 
when pushed to the utmost logical limits, metageometry 
proves not only futile, but emphasizes the need for 
a sharp turning of the path of search from the in- 
tellectual or material to the spiritual or intuitional. 
Indeed, it becomes painfully certain that the Golden 



x PREFACE 

Fleece of profounder knowledge will be discovered 
never by an expedition whose bark has its sails set 
for the winds of mathematical seas. But, contrarily, 
a new bark, moored at the furthermost shore of the 
sea of intellectuality with sails set for the winds which 
come from the realm of intuitional perception, must 
be seized. Whereupon, by the straightest line, we 
shall, at the last, land upon the shore of realism, of 
truth all inclusive. 

Mathematical evidences have been used in these 
dkcussions because they, of all lines of knowledge, 
afford a more just exemplification of intellectual evo- 
lution. The science of mathematics is the measure of 
the quality of intellectual growth and, therefore, its 
data, its postulates, hypotheses and advances clearly 
mark the stages of the intellectual movement. 

Chapter X is the natural and logical sequence of 
the inquiry into the question of spatiality. The con- 
clusions reached therein and the obvious inferences 
which should be drawn from the arguments presented 
flow inevitably not only from the evidences of mathe- 
matical data but of the common observations of life. 
And while we disclaim any intention of demanding 
acceptance of them as final, authoritative declarations, 
we shall be satisfied if the readers of this volume be 
incited to solve for themselves the problems which 
these queries naturally suggest. Happy indeed shall 
be the outcome if there be any who, following the path 
sketched herein, shall find the solution of the Mys- 
tery of Space and apply its meanings to the enhance- 
ment of the values of the intuitive life. 

In conclusion, the author esteems it a special obli- 
gation of gratitude that he should here acknowledge 



PREFACE xi 

the debt which he owes to all of his friends who have 
in any way assisted or encouraged him in the com- 
pletion of this work. Among those whom he is per- 
mitted to thank in this way is Mr. James Rindfleisch 
who, having very kindly prepared the illustrations for 
the photo-engravers, is deserving of special mention. 

Robert T. Browne. 

New York City, 
1919. 



CONTENTS 



PAGE 

Preface vii 

Introduction : Explanatory Notes i 



PART ONE 

CHAPTER I 

The Prologue 

On the Variability of Psychic Powers — The Discovery of 
the Fourth Dimension Marks a Distinct Stage in 
Psychogenesis — The non-Methodical Character of 
Discoveries — The Three Periods of Psychogenetic 
Development — The Scope and Permissibility of 
Mathetic License — Kosmic Unitariness Underlying 
Diversity 23 

CHAPTER II 

Historical Sketch of the Hyperspace Movement 

Egypt the Birthplace of Geometry — Precursors: Nasir- 
Eddin, Christoph Clavius, Saccheri, Lambert, La 
Grange, Kant — Influence of the Mecanique Analy- 
tique — The Parallel Postulate the Root and Substance 
of the Non-Euclidean Geometry — The Three Great 
Periods: The Formative, Determinative and Elabo- 
rative — Riemann and the Properties of Analytic 

Spaces 44 

xiii 



xiv CONTENTS 

CHAPTER III 
Essentials of the Non-Euclidean Geometry 

PAGE 

The Non-Euclidean Geometry Concerned with Concep- 
tual Space Entirely — Outcome of Failures at Solving 
the Parallel-Postulate — The Basis of the Non- 
Euclidean Geometry — Space Curvature and Mani- 
foldness — Some Elements of the Non-Euclidean 
Geometry — Certainty, Necessity and Universality as 
Bulwarks of Geometry — Some Consequences of Efforts 
at Solving the Parallel-Postulate — The Final Issue 
of the Non-Euclidean Geometry — Extended Con- 
sciousness 69 

CHAPTER IV 

Dimensionality 

Arbitrary Character of Dimensionality — Various Defini- 
tions of Dimension — Real Space and Geometric Space 
Differentiated — The Finity of Space — Difference Be- 
tween the Purely Formal and the Actual — Space as 
Dynamic Appearance— The A Priori and the A 
Posteriori as Defined by Paul Cams 92 

CHAPTER V 

The Fourth Dimension 

The Ideal and the Representative Nature of Objects in 
the Sensible World — The Fluxional, the Basis of 
Mental Differences — Natural Symbols and Artificial 
Symbols — Use of Analogies to Prove the Existence of 
a Fourth Dimension — The Generation of a Hyper- 
cube or Tesseract — Possibilities in the World of the 
Fourth Dimension — Some Logical Difficulties Inher- 
ing in the Four-Space Conception — The Fallacy of 
the Plane-Rotation Hypothesis — C. H. Hinton and 
Major Ellis on the Fourth Dimension 118 



CONTENTS xv 



PART TWO 

spatiality: an inquiry into the essential nature 
of space as distinguished from the mathe- 
matical interpretation 

CHAPTER VI 
Consciousness the Norm of Space Determinations 

PAGE 

Realism Is Determined by Awareness — Succession of De- 
grees of Realism — Sufficiency of Tridimensionality — 
The Insufficiency of Self-consistency as a Norm of 
Truth — General Forward Movement in the Evolu- 
tion of Consciousness Implied in the Hyperspace Con- 
cept — The Hypothetical Nature of Our Knowledge — 
Hyperspace the Symbol of a More Extensive Realm 
of Awareness — Variations in the Methods of Inter- 
preting Intellectual Notions — The Tuitional and the 
Intuitional Faculties — The Illusionary Character of 
the Phenomenal — Consciousness and the Degrees of 
Realism 161 

CHAPTER VII 

The Genesis and Nature of Space 

Symbology of Mathematical Knowledge — Manifestation 
and Non-manifestation Defined — The Pyknon and 
Pyknosis — The Kosmic Engenderment of Space — On 
the Consubstantiality of Spatiality, Intellectuality, 
Materiality, Vitality and Kosmic Geometrism — 
Chaos-Theos-Kosmos — Chaogeny and Chaomor- 
phogeny — N. Malebranche on God and the World — 
The Space-Mind — Space and Mind Are One — The 
Kosmic Pentoglyph 203 



xvi CONTENTS 

CHAPTER VIII 
The Mystery of Space 

PAGB 

The Thinker and the Ego — Increscent Automatism of 
the Intellect — The Egopsyche and the Omnipsyche — 
Kosmic Order or Geometrism — Life as Engendering 
Element — The Mystery of Space Stated — Kathekos 
and Kathekotic Consciousness — Function of the Ideal 
— The Path of Search for an Understanding of the 
Nature and Extent of Space Must Proceed in an In- 
verse Direction 242 

CHAPTER IX 

Metageometrical Near-truths 

Realism Is Psychological and Vital — The Impermanence of 
Facts — On the Tendency of the Intellect to Frag- 
mentate — The Intellect and Logic — The Passage of 
Space — Kosmometer and Zoometer, Instruments for 
the Measurement of the Passage of Space and the 
Flow of Life — The Disposal of Life and the Power 
to Create — Space, a Dynamic, Creative Process — 
Numbers and Kosmogony — Kosmic Significance of 
the Circle and the Pi-proportion — Mechanical Tend- 
ence of the Intellect and Its Inaptitude for the Under- 
standing of Life — The Criterion of Truth 284 

CHAPTER X 

Media of New Perceptive Faculties 

The Spiritualization of Matter Is the End of Evolution — 
Sequence and Design in the Evolution of Human 
Faculties — The Upspringing Intuition — Evidences of 
Supernormal Powers of Perception and the Possibility 
of Attainment — The Influence and Place of the Pitui- 
tary Body and the Pineal Gland in the Evolution of 
Additional Faculties — The Skeptical Attitude of Em- 
pirical Science and the Need for a More Liberal 
Posture — The General Results of Pituitarial Awaken- 
ing Upon Man and the Theory of Knowledge 327 

Bibliography 359 

Index 367 



THE 
MYSTERY OF SPACE 



THE MYSTERY OF SPACE 



INTRODUCTION 

Explanatory Notes 

The following interpretation of words, phrases 
and notions occurring in the text, and also biographical 
sketches which the scope and purpose of the book itself 
make it impracticable to elaborate, are appended with 
the view to facilitating its perusal. 

AT-ONE-MENT (state of unity, unitariness) ; denotes 
the ultimate state of oneness towards which all evolutionary 
movement tends; applied to consciousness, indicates the final 
expansion of consciousness wherein it coincides with the uni- 
versal consciousness in extent and quality of comprehension. As 
applied to things, denotes the unification of all movements, 
tendencies, and evolutions as a singularity; the end of all evolu- 
tionary activity (vide p. 270). 

BELTRAMI, EUGENIO, was born at Cremona, No- 
vember 16, 1835; there he attended the elementary schools, the 
gymnasium and the lyceum, excepting the scholastic year 
1848-49 when he was at the Gymnasium of Venice, now known 
as Marco Polo. He finished his lyceal studies in the summer of 
1853, and in the following autumn (November) became a 
student in the Mathematical Faculty of the University of 
Pavia, after having obtained a scholarship there on the Casti- 
glioni Foundation in the Collegio Ghisleri. 



2 THE MYSTERY OF SPACE 

In 1854, the succeeding year, he was expelled from this 
college in company with five of his colleagues who were ac- 
cused of promoting "disorders" against the Abbot Leonardi, 
rector of the college. The expulsion brought him many hard- 
ships and disappointments, and for two years he drifted along 
merely existing as his family was too poor to have him matricu- 
lated at another university. But in 1856, he went to Verona 
where he succeeded in securing employment as secretary to the 
engineer, Diday, in the Government service of Lombardy- 
Venice. On January 10, 1857, he was dismissed from this 
position "for political reasons"; but as the annexation of Lom- 
bardy to Piedmont occurred soon thereafter, he became again 
attached to the office of Diday, his former employer, when it 
was transferred to Milan as a consequence of political changes. 

At Milan Beltrami took up his mathematical education in 
real earnest as he now had access to Professor Brioschi, his 
former tutor, and also Luigi Cremona. Through the influence 
of these two men he was designated (October 18, 1862) "Pro- 
fessore straordinario" in the University of Bologna. His work 
on Surfaces of Constant Negative Curvature, as the pseu- 
dosphere, and his application of the expression given by Lo- 
bachevski (q.v.) for the angle of parallelism, very definitely 
secure for him a place among the foremost workers in the field 
of the non-Euclidean geometry. He postulated a theorem, 
known as Beltrami s Theorem, which he stated as follows: 
"The center of a circle circumscribing a triangle is the center 
of gravity of the centers of its inscribed and escribed circles." 
He died in the year 1900. (Vide Amer. Math. Mo., Vol. IX, 
P. 59.) 

BOLYAI, JANOS (1802-1860), was born at Kleansen- 
burg, Hungary. He is said to have inherited his mathematical 
genius from his father, Bolyai Farkas (1775-1856), who was 
born at Bolya, Hungary. Being a very spirited youth, his 
progress in his studies was most remarkable. He completed 
the curriculum at the Latin school when only twelve years of 



INTRODUCTION 3 

age. Was graduated from the Philosophical Curriculum as a 
result of two years of study and then entered the Viennese 
Academy of Engineers. Was appointed lieutenant at Temes- 
varlin, 1823, whence on November 3, 1823, he wrote his 
father: "I have discovered such magnificent things that I am 
myself astonished at them. It would be damage eternal if 
they were lost. When you see them, my father, you yourself 
will acknowledge it. Now, I cannot say more, only so much: 
that from nothing I have created another wholly new world." 
This letter was written in the Magyar language and has been 
preserved at the Marcos Vasarhely, Hungary. The mathemati- 
cal conceptions formulated by him became the appendix of 
the Tentamen, a book which his father had written on the 
Theory of Parallels. 

His Science Absolute of Space was translated into the 
French in 1868 by the French mathematician, J. Houel, to 
whom belongs the credit of popularizing the works both of 
Bolyai and Lobachevski. (Vide Science, n. s., Vol. 35, 
No. 906, 1912.) 

CAYLEY, ARTHUR, born at Richmond, Surrey, Eng- 
land, August 16, 1 821; studied at King's College school; 
entered Trinity College, Cambridge, already a well equipped 
mathematician at the age of seventeen. When but twenty-one 
years of age he took two of the highest honors in the Uni- 
versity of Cambridge. He was Senior Wrangler and First 
Smith's Prizeman. He published his first paper in 1 841 and 
this was followed by eight hundred memoirs. 

For fourteen years he practiced as Conveyancer. In 1863 
Lady Sadler's various trusts were consolidated, and a new 
Sadlerian professorship of Pure Mathematics was created for 
the express purpose of affording a place for Cayley. Mean- 
while, as early as 1852 he was a fellow of the Royal Society; 
in 1858 he joined Sylvester and Stokes in publishing the 
Quarterly Journal of Pure and Applied Mathematics. 

He was for a considerable time principal adviser as to the 



4 THE MYSTERY OF SPACE 

merits of all mathematical papers which were presented for 
publication to the Royal Society, the Astronomical Society, the 
Mathematical Society and the Cambridge Philosophical Society. 
He is said to have been the "most learned and erudite of mathe- 
maticians," and much of the material, therefore, which now 
constitutes the basis of the non-Euclidean geometry is due to 
his laborious efforts and comprehensive knowledge of mathe- 
matics. (Vide Review of Reviews, Vol. II, 1895, Sketch, re- 
printed from Monist.) 

CHAOGENY (Gr. Chaos, disorder, geny — generating, 
evolution) ; the evolution of chaos into order. A kosmic proc- 
ess involving the elaboration of the original, formless world- 
plasm into the first faint signs of orderliness; the beginnings 
of the movement of life or the Creative Logos in preparation 
of the field of evolution. 

CHAOMORPHOGENY (Gr. Chaos, disorder— Morphe, 
form — geny, becoming, generating) ; evolution of the space- 
form, the universe; the establishment of the metes and bounds 
of the universe; also, the origination and characterization of all 
forms as to tendence, purpose and limitations. 

CONCEPTUALIZATION— The act of conceptualizing, 
the formulation of concepts; the process by which the Thinker 
arrives at concepts; the logical procedure by which the con- 
sistency of a scheme of thought is established. 

CONSTRUCTION, IDEAI^-A purely formal concep- 
tion; a theory, hypothesis; a logical determination not neces- 
sarily based upon facts, but possessing virtue because of con- 
sistency; a self-consistent scheme of thought. 

COSMOS — Whenever the term "cosmos" appears in the 
text spelled as here shown, it refers to phenomena pertaining to 
the earth or the solar system; when spelled "kosmos" reference 
is made to the universe as a whole. 



INTRODUCTION 5 

CRITERION OF TRUTH— Defined in the text as a 
four-fold standard of reference, embracing the following ele- 
ments, namely, the causal, the sustentative, relational and de- 
velopmental. Lacking any one of these, no view of truth is 
more than fragmentary. Applied to space, it contemplates an 
inquiry into the genesis or causal aspect, an accounting for the 
duration aspect, a recognition of its relation to the totality of 
objects, and lastly, a prophecy of its telestic or perfective cul- 
mination. This test has been applied to the study of space as 
sketched in the text and the conclusions reached are an out- 
come of the inquiry directed along these lines. 

CURVATURE OF SPACE— A doctrine formulated by 
Riemann and which maintains that space is curved, and con- 
sequently, all lines drawn therein are curved lines. Pro- 
fessor Pickering aptly describes the results of movements in 
a curved space by pointing out that if we go far enough east 
we arrive at the west; north, we arrive at the south; towards 
the zenith, we arrive at the nadir, and vice versa. 

DEI FORM — The basic idea indicated is that the universe 
is the form or body of the supreme deity, since He is not only 
immanent in the kosmos, but sustains it by His life; that in 
order to create a manifested universe, it was necessary to limit, 
or sacrifice, in a measure, His own illimitability. Viewed in 
this light, the kosmos assumes an added significance. 

DIMENSION — (L. Dimensio, to measure), measure- 
ment; a system of space measurement. The Euclidean geome- 
try recognizes three dimensions or coordinates as being necessary 
to establish a point position; witness, the corners of a cube to 
form which three of the edges come together at a point. These 
edges represent coordinates. For the purposes of metageometry, 
the term dimension has been variously defined, as, direction, 
extent, a system of space measurement, or a system of coordi- 
nates. Regarded as a series of coordinates, it became possible 
to postulate a system which required four coordinates to estab- 



6 THE MYSTERY OF SPACE 

lish the position of a point, as in the hypercube. There may 
be five, six, seven, eight, or any number of such coordinate sys- 
tems according to the kind of space involved in the calculations. 
Determinations based upon the logical necessities of the vari- 
ous coordinate systems have been found to be self-consistent 
throughout and, therefore, valid for metageometrical purposes. 
Much depends upon the definition; for, after the definition has 
been once determined it remains then merely to make inferences 
and conclusions conform to the intent of the definition. 

DIVERSITY— Philosophically, the idea indicated has ref- 
erence to all dissimilarities, differences, inequalities, divergent 
tendencies, movements and characteristics to be noted in the 
universum of life; the antithesis of kosmic unity; the natural 
outcome of life in seeking expression; the result of the frag- 
mentative tendency of life. 

DUODIM (duo, two; dim, abbreviation of dimen- 
sion) — A hypothetical being supposed to be possessed with a 
consciousness adapted only to two dimensions; a dweller in 
"Flatland" or two-space whose scope of motility is limited to 
two directions, as on the face of a plane; a term invented by 
hyperspace advocates for the purpose of establishing by analogy 
some of the characteristics of the four space and also its 
rationality. 

DUOPYKNON (duo, two— pyknon, primary unit in the 
process of kosmic involution, a condensation) — Secondary 
phase in the elaboration of chaos into kosmic order. DUOPYK- 
NOSIS (duo, two, secondary — pyknosis, process of condensation 
and origination) — The second period of the involutionary 
movement of life during which the duadic plane of the kosmos 
is being established; the second, in the series of seven distinct 
phases, of space-genesis; dual differentiation of kosmic plasm. 
Duopyknosis contemplates that, in the passage of the kosmos- 
to-be from the plane of non-manifestation to the plane of mani- 
festation, there are seven distinct, though interdependent and 



INTRODUCTION 7 

interrelation^, stages through which life passes, and that, of 
these, it is the second. It relates to the plane of non-manifesta- 
tion and is, therefore, beyond the ken of the intellectuality, 
being a symbol. 

EGOPSYCHE (Ego, the self-conscious I— psyche, soul)— 
The mental, emotional and physical mechanisms of man, the 
Thinker. These include the purely mental system, the emo- 
tional or affective mechanism, the nerve-systems (cerebro-spinal 
and sympathetic) and the brain; the objective or sense-derived 
consciousness of the Thinker which is elaborated from the 
total mass of perceptions transmitted through the senses; the 
medium of self-consciousness; the intellectual consciousness as 
distinguished from the intuitional or omnipsychic conscious- 
ness (q.v.). 

The notion of the egopsychic consciousness is based upon 
data already empirically determined from the mass of evi- 
dences everywhere observable. It seems to be apparent that 
there is a consciousness, a seat of knowledge, in man the con- 
tent of which is unknown to the sense-consciousness. Dreams, 
premonitions, intuitions, impressions and the totality of all such 
phenomena substantiate this view. Furthermore, it is agreed 
that the source of the intuition is not identical with that of 
the intellect. The egopsychic consciousness, accordingly, is 
purely intellectual. 

FLUXION, PSYCHIC— The difference between a mental 
image and an object; an image is the representation of cer- 
tain salient or cardinal characteristics of an object, sufficient 
for identification; but an image is not congruent, in every re- 
spect, with the object. Thus when we perceive an object, 
although as Bergson contends we perceive it in the place where 
it is and not in the brain, it is the image of the object which 
takes its place in memory and not the object itself. There is, 
of course, a marked disparity between this memory-image and 
the object. Even if the image possessed one of the properties 



8 THE MYSTERY OF SPACE 

of the object, as, size, it could not take its place in memory, 
and neither could it do so if it possessed any of the properties 
of the real object. Consciousness is such that all due allow- 
ances are made for these conditions and the mind is able to 
retain more or less exact knowledge of these properties in the 
image; but there is a difference, small though it may be. This 
difference is the psychic fluxional. 

FOHAT (Skt) — A term applied to the Creative Logos 
who is said to be the generating element in the differentiation 
of chaos into kosmic orderliness; the supreme deity in the role 
of Creator. 

FORM, PURE — An abstraction arrived at by subtract- 
ing the last vestige of materiality or substantiality from an 
idea and viewing the remains as a pure unsubstantial form or 
idealization; the shell or frame-work of a material object or 
condition; existing in idea or thought only; a mental concep- 
tion regarded as a type or norm; a purely hypothetical con- 
struction. 

FOUR SPACE— Often referred to as the fourth dimen- 
sion (vide Chapter V) ; a space in which four coordinates (four 
lines drawn perpendicular to each other) are necessary and 
sufficient to establish the position of a point, as, a hypercube. 

GAUSS, CHARLES FREDERICK, born at Brunswick, 
April 30, 1777. His father, being a bricklayer, had intended 
that he should follow the same occupation. So, in 1784, 
Charles was sent to the Burner Public School in Brunswick, in 
order that he might be taught the ordinary elements of educa- 
tion. But during his attendance at this school, his unusual in- 
telligence and aptitude attracted the attention and friendship 
of Professor Bartels who later became the Professor of Mathe- 
matics at Dorpat. In 1792, through the kindly representa- 
tions of Professor Bartels to the Duke of Brunswick, young 



INTRODUCTION 9 

Gauss was sent to the Collegium Carolinum. This greatly dis- 
pleased his father as he saw in this move the frustration of his 
plans for Charles. In 1794, however, Gauss entered the Uni- 
versity of Gottingen still undecided whether he should make 
mathematics or philology his life work. While residing at 
Gottingen, he made his celebrated discoveries in analysis and 
these turned his attention definitely to the field of mathematics. 

He completed his studies at Gottingen and returned to 
Brunswick in 1798, residing at Helmstadt where he had access 
to the Library in the preparation of his Disquisitiones Arith- 
meticae which was published in 1801. He received his doc- 
torate degree (Ph.D.) on July 16, 1799. His next notable 
work was the invention of a method by which he calculated 
the elements of the orbit of the planet Ceres which had been 
discovered by Piazzi, January 1, 1800, and who had left no 
record of his calculations by which other astronomers could 
locate the planet. Gauss also calculated an ephemeris of 
Ceres* motion by means of which De Zach rediscovered the 
planet December 31, 1800. 

His Theoria Motus Corporum Coelestium in Conicis Sec- 
tionibus Solem Ambientium, in which the author gives a "com- 
plete system of formulae and processes for computing the move- 
ments of a heavenly body revolving in a conic section" is an 
outgrowth of his early researches and brought him lasting fame. 

Through the influence of his friend Olbers, he was ap- 
pointed, July 9, 1807, first director of the new Gottingen 
Observatory, and Professor of Astronomy in the University, a 
position which he held until the end of his life. He died 
February 23, 1855. (Vide Astronomical Society Notices, Vol. 
16, p. 80, 1856; also Nature, Vol. XV, pp. 533"537> 1877.) 

GEOMETRISM— Of geometrical quality; a notion de- 
rived from Plato's declaration: "God geometrizes." It was 
his belief that the creative acts of the deity are executed in 
accordance with geometric design and laws; that in the totality 
of such acts there necessarily inheres a latent geometric quality. 



io THE MYSTERY OF SPACE 

Kant closely adhered to this notion in his discussions of space 
as an aspect of divine intelligence. He believed that the in- 
tellect merely rediscovers this latent geometrism when it turns 
to the study of materiality, and this belief is shared by BERGSON, 
the foremost metaphysician of the present time. 

HYPERSPACE (hyper, above, beyond, transcending — 
space) — That species of space constructed by the intellect for 
convenience of measurement ; an idealized construction ; a purely 
arbitrary, conventional mathematical determination; the fourth 
dimension; any space that requires more than three coordinates 
to fix a point position in it, as, a five space, an w-space. 

INTUITOGRAPH— The means by which the omni- 
psychic consciousness transmits intuitional impressions to the 
egopsychic or intellectual consciousness. An intuitogram is a 
direct cognition, an intuition; a primary truth projected into 
the egopsychic consciousness by the Thinker. It is recognized 
that, under the necessities of the present schematism of things, 
it is exceedingly difficult to propagate an intuition, especially 
with the same degree of ease as concepts are propagable; yet, 
this is believed to be a condition which will be overcome as the 
evolution of the higher faculties proceeds. 

INVOLUTION — Process of enfolding, involving; anti- 
thesis of evolution; philosophically, the doctrine of involution 
maintains that, during the process of kosmic pyknosis (space- 
genesis), all that is to be expressed, developed and perfected 
as a result of the evolutionary movement was first involved, 
enfolded or deposited as latent archetypal tendencies and radi- 
cles in the original world-plasm; that, as the involutionary 
movement proceeded through the various phases of space- 
genesis, these became more and more phenomenal until at last 
they terminated in the elaboration of a manifest universe: each 
stage, accordingly, of the involutionary procedure became the 
basic substructure of a plane of specialized substance or ma- 



INTRODUCTION 11 

teriality and consciousness. Thus it 2ppears that evolution 
really begins where involution ends (vide Fig. 18), and the 
two opposing processes constitute the dualism of life as gen- 
erating element. This notion has been symbolized in the 
Lingam yoni of Hellenistic philosophies, also in Yang and Yin 
of Chinese philosophy, which represent the original pair of 
opposites. 

KATHEKOS — A purely arbitrary term devised for the 
express purpose of providing a convenient symbol to convey the 
idea embodied in the triglyph, Chaos-Theos-Kosmos, and is 
composed of the first three letters in each one of the terms of 
the triglyph; hence, symbolizes the triunity and interaction in- 
volved in the resolution of chaos into an orderly kosmos by 
the will of the Creative Logos. Thus, "kathekos" embodies a 
quadruplicate notion, namely, chaos, Creative Logos, manifested 
kosmos, and the creational activity of the Logos in the trans- 
mutation of disorder into order. The justification for this term, 
therefore, resides in its convenience, brevity and comprehen- 
siveness. 

By referring to figure 17, it will be seen that Kathekos 
divides into two kinds — involutionary, or that which pertains to 
involution, and evolutionary, or that that pertains to evolution. 
It thus comprises the beginning and the end of the world age 
or cycle and pertains to non-manifestation. The raison d'etre 
of this differentiation is embodied in the notion that, on the in- 
volutionary arc of the cycle, the chaogenic period represents a 
phase of the world age when space-genesis is in an archetypal 
state wherein are involved all possibilities that are to become 
manifest in the kosmos, and on the evolutionary arc, the kathe- 
kotic period which is parallel to the chaogenic and represents a 
phase of the world age when the kosmos has reached ultimate 
perfection, embodying the perfected results of the possibilities 
which inhered in the chaogenic period or in involutionary 
kathekos. Thus, kathekos is dual in nature, on the one hand 
representing kosmic potency, and on the other, kosmic perfec- 



12 THE MYSTERY OF SPACE 

tion of these potencies. It is Alpha, as related to involution, 
and Omega, as related to evolution. 

KATHEKOSITY — A derivative, signifying creative ac- 
tivity and all that it implies; the state of consciousness or cogni- 
tion corresponding thereto. 

KLEIN, FELIX (1849—), born at Dusseldorf; studied 
at Bonn, and when only seventeen years of age was made as- 
sistant to the noted Plucker in the Physical Institute. He 
took his doctorate degree in 1868, then went to Berlin, and 
later to Gottingen where he assisted in editing Plucker's 
works. He entered the Gottingen faculty in 1871 ; became Pro- 
fessor of Mathematics at Erlangen in 1872; and subsequently 
held professorships at Munich, 1875; Leipzig, 1880, and Got- 
tingen, 1886. No one else in Germany has exerted so great 
influence upon American mathematics as he. 

KOSMOS— See Cosmos. 

LA GRANGE, JOSEPH LOUIS, born at Turin, January 
25, 1736; died at Paris, April 10, 1813; regarded as the great- 
est mathematician since the time of Newton. It may be in- 
teresting to note that La Grange remarked that mechanics is 
really a branch of pure mathematics analogous to a geometry 
of four dimensions, namely, time, and the three coordinates of 
the point in space. (Vide Ball's Account of the History of 
Mathematics. ) 

LIE, SOPHUS, a noted mathematician, referred to as 
the "great comparative anatomist of geometric theories, creator 
of the doctrines of Contact Transformations and Infinite Con- 
tinuous Groups, and revolutionizer of the Theory of Differen- 
tial Equations." 

LOGOS — The supreme deity of the phenomenal universe; 
Creator ; Fohat ; a planetary god ; the deity of a solar system. 



INTRODUCTION 13 

MANVANTARA (Skt.)— A world age; the periods of 
involution and evolution combined; the stage during which 
the universe is in manifestation; a Day of Brahma. 

MATHESIS (Gr. mathein, to learn)— Erudition; pro- 
found learning; the realm of metaphysical conceptions; the 
field of higher mathematics; the sphere of conceivability ; the 
theoretical. 

MENTOGRAPH — A cognitive factor consisting of a 
complete perception fused or in coalescence with a memory- 
image. Pure memory, of itself, is without utility as an aid to 
cognition; but, when nourished or supplemented by the sub- 
stance of perception it becomes the basis of intellectual con- 
sciousness. 

METAGEOMETRY (Gr. Meta, beyond, transcending 
— geometry) — Commonly, any kind of geometry that differs 
from the Euclidean, as the non-Euclidean; a geometry based 
upon the assumption that the angular sum of a triangle is 
greater or less than two right angles; the highest form of 
geometry; a system of idealized mathematical constructions. 
Sometimes called "pangeometry" ; designated by Gauss as 
"Astral Geometry"; the geometry of hyperspace. It consists 
of results arrived at by geometers in seeking a proof of the 
parallel-postulate. 

META-SELF — The higher self in man ; the universal self ; 
the one self of which all individual selves are but fragments or 
parts. In man, it is coordinate with the omnipsyche (q.v.) 
and as such is the medium of kosmic consciousness. 

MORPHOGENY (Gr. Morphe, form, vehicle, body— 
geny, evolution) — The evolution of forms, the production of 
individual bodies or vehicles for life, including organs and 



i 4 THE MYSTERY OF SPACE 

faculties. Morphogenic — a derivative; pertaining to mor- 
phogeny; a kosmic process (vide figs. 17 and 18). 

^-DIMENSIONALITY— Quality of conceptual space by 
virtue of which it may be regarded as possessing an indefinite 
number of dimensions. 

NEAR-TRUTH — Any statement or view which is based 
upon partial knowledge; predicates concerning a class or genus 
derived from limited acquaintance with particulars of the class 
or genus; statements based upon logical determinations inher- 
ing in idealized constructions and applied to concrete or objec- 
tive conditions; an abstraction viewed as a reality; the applica- 
tion of the qualities of abstractions to realities. 

NEUROGRAM — Psychologically, a movement received by 
the afferent nerves in the form of a stimulation and transmitted 
through the brain and efferent nerves as either a reflex or vol- 
untary action; a nerve impulse; a perception; a primary unit 
of intellectual consciousness; cf. Intuitogram. 

NEWCOMB, SIMON (1835-1909), born at Wallace, 
Nova Scotia; educated in his father's school and came to the 
United States in 1853. Began, in 1854, teaching in Marj'- 
land; was appointed computer on Nautical Almanac at Cam- 
bridge in 1857; was graduated at Lawrence Scientific School 
in 1858; appointed Professor of Mathematics in the U. S. Navy 
in 1 86 1. He supervised the construction of the 26-inch 
equatorial telescope at the Naval Observatory, and was secre- 
tary of the Transit of Venus Commission; was a member of 
nearly all of the Imperial and Royal societies of Europe and 
of the various societies in the United States, receiving the 
Copley Medal in 1874; the Huygens, 1878; the Royal Society, 
1890, and the Bruce Medal in 1898; held the presidency of 
the following learned societies, viz: American Association for 
the Advancement of Science, 1877; Society for Psychical Re- 



INTRODUCTION 15 

search, 1 885-1 886; American Mathematical Society, 1897- 
1898; the Astronomical and Astrophysical Society of America 
from its foundation in 1899. He rendered notable service in 
popularizing the doctrine of hyperspace. 

NORM — An authoritative standard; model or type; stand- 
ard of reference. The choice of a norm for spatial determina- 
tions cannot abide in any premise except that which naturally, 
and not artificially and conventionally, conforms to what is 
actually perceived; if so, there should be justification for chal- 
lenging the wisdom and utility of the present schematism of 
things. There is an inherent conformity of space with intellect 
and intellect with space, and because of this natural comple- 
mentarity of part with part and whole with whole, space cannot 
be otherwise than the intellectuality normally conceives it to 
be, provided, of course, that the cognitive movement is free 
and untrammeled by arbitrary hindrances. Consciousness, 
therefore, is the norm or standard of reference for all questions 
arising out of a consideration of spatiality. 

OMNIPSYCHE— A term used to denote the Thinker's 
cognitive apparatus; the universal soul manifesting in indi- 
viduals; the consciousness of the Thinker in virtue of which 
he is at-one with the universal consciousness; the medium of 
kosmic consciousness ; the source of the intuition, cf. Egopsyche. 
The divinity in man (which is taken for granted), or his 
highest self can in no way be said justly to take its rise from 
sense-experience or from any bodily process. If divine, then 
eternal, and therefore, persistent. Broadly, the doctrine of 
evolution recognizes the passage of life from form to form, 
adding a little to each successive form and inevitably pushing 
each to a higher degree of perfection. Now, what is it that 
passes from form to form? Is it undifferentiated life or is it a 
specialized form of life? From every evidence, it would be 
judged that the life that ensouls an individual form is a special- 
ized principle, i.e., limited to the execution of a given purpose. 



X6 THE MYSTERY OF SPACE 

If life as a specialized principle, limited to the execution of a 
given purpose in each form, passes on, it must preserve, at least, 
the sublimated results obtained during its residence in each in- 
dividual form. It would thus become a sort of reservoir con- 
taining all these transmuted results. The omnipsyche, within 
the meaning of the text, is precisely this specialized life principle. 

PARALLEL-POSTULATE— Variously referred to as the 
Xlth, Xllth and XHIth axiom of the Elements of Euclid; 
stated by Manning, in Non-Euclidean Geometry, p. 91, in the 
following form: "If two lines are cut by a third, and the sum 
of the interior angles on the same side of the cutting line is 
less than two right angles, the lines will meet on that side 
when sufficiently produced." This celebrated postulate has 
proven to be the most fruitful ever devised; for it embodies 
in itself the possibility of three geometries based respectively 
upon the following assumptions, namely: I. That there exists 
a triangle, the sum of whose angles is congruent to a straight 
angle, the Euclidean; II. That there exists a triangle the sum 
of whose angles is less than a straight angle, the Lobachevsk- 
Ian; III. That there exists a triangle the sum of whose angles 
is greater than a straight angle, the Cayley-Klein. Speaking 
of the content of the last two named, Edward Moffat 
Weyer, 1 says: "Hypothetical realms, wherein the dimensions 
of space are assumed to be greater in number than three, yield 
strange geometries, which are only card castles, products of a 
sort of intellectual play in the construction of which the laws 
of logic supply the rules of the game. The character of each is 
determined by whatsoever assumption its builder lays down 
at the start." 

PASSAGE OF SPACE— A phrase connoting the move- 
ment of space from chaos to perfect order, a process believed 
to be infinite. The genesis of space necessarily implies an 

^ide Popular Science Monthly, vol. 78, p. 554, 1911. 



INTRODUCTION 17 

elaboration, a procedure, by which the metamorphosis from dis- 
order to kosmic order is made, and this movement is referred 
to as the "passage of space," a phenomenon thought to be 
measurable by means of a suitable instrumentality. 

PERISOPHISM— See Near-Truth, 

PSEUDOSPHERE — A surface of constant negative cur- 
vature; basis of Beltrami's metageometrical calculations; sur- 
face resembling a champagne glass or common spool. The 
assumption that space is pseudospherical has given rise to the 
notion of space-curvature and various other conceptions. 

PSYCHOGENY (Gr. Psyche, Soul— geny)— History of 
the evolution of the soul or the development of the senso- 
mechanism in organisms. Ernst Haeckel has traced the 
psychogeny of man through twenty-two different stages from 
the moneron to the anthropoid apes, and man. 

PRALAYA (Skt.) — Kosmic quietude; the period during 
which the universe is not in manifestation; gestatory period; 
kosmic inactivity; opposed to manvantara (q.v.) ; figuratively, 
the kosmic womb; world egg. 

PYKNON (Gr. pyknon, hard) — The principle of kosmic 
condensation; the primary basis of space-genesis; the initiation 
of the process by virtue of which chaos is elaborated into kosmic 
order. PYKNOSIS — The process of spatial engenderment. 
There are seven of these processes, each indicating a phase of 
duration, namely: MONOPYKNOSIS, the primary phase; 
DUOPYKNOSIS, secondary; TRIPYKNOSIS, tertiary. 
These three pertain to the plane of non-manifestation, the pra- 
layic or gestatory duration-phase. The results arrived at dur- 
ing these duration-phases are concentrated in the Quartopyknotic 
which corresponds to the causal plane of manifestation or pure 
kosmic spirituality. (JUINTOPYKNOSIS, a process con- 



18 THE MYSTERY OF SPACE 

cerned in the genesis of mentality; SEXTOPYKNOSIS, kosmic 
sensibility; SEPTOPYKNOSIS, kosmic materiality. These 
seven phases of duration constitute the scope of space genesis or 
kosmogenesis, and incidentally depose the substructure of kos- 
mic materiality, sensibility, intellectuality and spirituality, as 
well as the higher trinity of kosmic modes. The ramifications 
of these principles are innumerable and omnipresent. (See 
Chapter VII.) 

QUARTODIM — A hypothetical being assumed to have a 
consciousness adapted to hyperspace or the fourth dimension, 
and whose scope of action is encompassed within a space which 
requires four coordinates, as the four-space. 

REALITY (Realism) — Life; the harmony existing among 
the parts to maintain their equilibrium in the whole; the prin- 
ciple of integrity subsisting among parts; kosmic vitality. 

RIEMANN, GEORGE FREDERICH BERNHARD, 
was born September 17, 1826, in the village of Breselenz, near 
Dannenburg, in Hanover. Until he was eight years of age 
his father was his sole tutor, but even at this age he exhibited 
great powers of arithmetical calculation. In the Spring of 1840 
young Riemann was sent to the Hanover Lyceum where he 
remained for two years, leaving in 1842 for the Gymnasium at 
Luneburg. Here, under the direction of Professor Schmal- 
FUSS, he learned very rapidly, and is said to have required only 
one week thoroughly to familiarize himself with Legendre's 
Theory of Numbers, 

On April 12, 1846 (Easter), he entered the University of 
Gottingen as a student of Theology in accordance with his 
father's wishes. His passion for mathematics, however, was 
so aroused by the lectures of Gauss that He begged his father 
to be allowed to devote himself entirely to the studies of his 
choice. For two years he studied under Jacobi at Berlin. He 
then returned to Gottingen, and was graduated, his thesis being 



INTRODUCTION 19 

a dissertation on the foundations of a general theory of func- 
tions of a variable complex magnitude. In 1854 ne qualified 
as a teacher by giving a lecture on the "Hypothesis on which 
Geometry is Founded." In 1857 he became "Professor Extraor- 
dinarius," and in 1859 was elected Corresponding Member 
of the Academy of Sciences of Berlin and in i860 a member of 
the Academy of Sciences of Gottingen. 

After four years of failing health, during which he visited 
Messina, Palermo, Naples, Rome, Florence, Pisa and Milan, 
he died at Lago Maggiore, July 20, 1866, in full possession of 
his faculties and conscious of his approaching end. 

SCHWEIKART, FERDINAND KARL (1780-1857), 
studied from 1796 to 1798 in Marburg, attending the mathe- 
matical lectures of J. K. F. Hauff. In 18 12 he became pro- 
fessor in Charkov, a position which he held for four years. In 
1816 he became a tutor in the City of Marburg where he re- 
mained until 1 820 when he transferred his labors to Konigsberg. 
It was during his tutorship at Charkov, Marburg and Konigs- 
berg that he, entirely alone and without the slightest suggestion 
from any man, developed and taught a non-Euclidean geometry 
to the students under his care. For copy of his treatise on 
non-Euclidean geometry, see Historical Sketch of the Hyper- 
space Movement, Chapter II. 

SCOPOGRAPHIC IMPRESSIONS— Sight perceptions 
fused with an associated memory-image, and forming the basis 
of action on external phenomena. 

SENSOGRAPHIC IMPRESSIONS— Perceptions or im- 
pulses transmitted through the nerves of a sense-organ; any 
impression acting through the media of the senses. 

SENSIBLE WORLD— The world of the senses; that 
which responds to the senses; the domain of perception; the 
phenomenal world; world of perceptual space. 



20 THE MYSTERY OF SPACE 

SPACE-CURVATURE (see Curvature of Space). 

SPACE-GENESIS — The process of spatial engenderment ; 
the movement of life as engendering agent in bringing into 
manifestation the kosmos; the story of the appearance of the 
organized kosmos. The genesis of space can only be symbolized, 
as has been done in the text, for the limitations of human con- 
sciousness do not otherwise admit of the empirical establish- 
ment of the notion of its detailed procedure. 

SPATIALITY — Space as a dynamic, creative movement; 
kosmic order, as opposed to disorder; the path of the engender- 
ing movement of life; the place of life. Spatiality, materiality, 
intellectuality and geometricity or the latent geometrism of the 
kosmos are thought of as being consubstantial and interde- 
pendent; but, of these, spatiality is regarded as the substance 
out of which the latter three are elaborated. 

SUPERCONCEPTUAL— The purely intuitional; an act 
of cognition performed without the detailed work of concep- 
tion derived from sense-data; conception of intuitions and their 
inter-relations; the Thinker's consciousness freed from intel- 
lectual characterization. 

SUPERPERCEPTION— Perception of conceptual rela- 
tions; a state of cognition wherein, instead of receiving percepts 
or images from the external world, then elaborating them into 
concepts, the Thinker apprehends composite images or con- 
cepts at first hand. It is a power which the liberated mind of 
the future will possess owing to the growing automatism of 
the intellect and the more facile expression of the intuitional 
consciousness. 

TESSERACT (Gr. Tessera, four, cube, tessella)— A 
hypercube (see Chapter V.) 



INTRODUCTION 21 

THINKER (Skt. Manu, thinker)— The real, spiritual 
man, as differentiated from his perceptive vehicles — mind, emo- 
tions and physical body; the omnipsychic intelligence who re- 
ceives, classifies, interprets and preserves percepts; the manipu- 
lator of concepts; in fine, the higher, spiritual man. 

The Thinker uses the various perceptive instrumentalities 
as so many tentacles or antennae by which he contacts the sen- 
sible world and makes the necessary adaptations to environ- 
ment. He is the pure intelligence which is the source of all 
cognitive motivation; opposed to ego, because the egopsychic 
instrumentality is essentially an individualizing, separative 
agency; while the Thinker's omnipsychic intelligence is the 
basis of his unity with the universal intelligence. This concep- 
tion of the Thinker implies that, as a spiritual intelligence, he 
is within and without the body, filling it as the ocean fills the 
sponge, encompassing, enveloping it and, at the same time, 
originating the totality of activities which manifest in and 
through the body. He is limited, therefore, in his manifesta- 
tions in the sensible world only by the pliability of his vehicles. 

TRANSFINITY— A state or condition that is incompre- 
hensible to finite intelligence; that which transcends the finite, 
yet is not infinite; less than infinity and greater than finity. 
Space is referred to as being transfinite rather than infinite in 
extent. But space transfinite should be distinguished from 
space "finite though unbounded." For, there would seem to 
be little worthy of choice between a "finite, unbounded space" 
and an infinite one. The absence of boundary would naturally 
suggest an infinite extent. And although Riemann who is 
the author of the "unbounded" space arbitrarily determined 
that such a space should be a manifold possessing a measure of 
curvature which could be determined either by counting or 
actual measurement, he undoubtedly knew, nevertheless, that 
while each manifold might be an "unbounded" space the totality 
of such manifolds, infinite in number, must also be infinite in 
extent. It would seem to do violence to common sense, if not 



22 THE MYSTERY OF SPACE 

to logical necessity, to view space both as "unbounded" and 
finite in extent, yet there would be no such difficulty in the 
recognition of space as being both transfinite and finite; be- 
cause it is conceivable that the extent and character of space 
finite should transcend a finite intellectuality, and yet not be 
infinite. 

TRIDIM — A being whose scope of consciousness is 
limited to a space of three dimensions, as ordinary human beings. 
TRIDIMENSIONALITY— That quality possessed by per- 
ceptual space by virtue of which it is necessary and sufficient to 
have three coordinates, and only three, to establish the position 
of a point. 

UNODIM — A hypothetical being assumed to have a con- 
sciousness limited to linear or one-space. 

ZONES OF AFFINITY— Regions in the domain of in- 
tellectuality wherein minds, possessing a common differential, 
rate of vibration or quality, adhere to certain tenets from 
choice. Schools of philosophy, religions, and all those major 
divisions of intellectual effort which divide and subdivide in- 
tellectual allegiance are believed to take their rise in this prop- 
erty of intellectuality in virtue of which all minds having a 
similar coefficient gravitate towards a common agreement, es- 
pecially where the movement is voluntary and untrammeled. 



PART ONE 

THE ESSENTIALS OF THE GEOMETRY OF 

HYPERSPACE AND THEIR 

SIGNIFICATIONS 



CHAPTER I 
The Prologue 

On the Variability of Psychic Powers — The Discovery of the 
Fourth Dimension Marks a Distinct Stage in Psycho- 
genesis — The Non-Methodical Character of Discoveries — 
The Three Periods of Psychogenetic Development — The 
Scope and Permissibility of Mathetic License — Kosmic 
Unitariness Underlying Diversity. 

In presenting this volume to the public profound 
apologies are made to the professional mathematician 
for the temerity which is shown thereby. All tech- 
nical discussion of the problems pertinent to the 
geometry of hyperspace, however, has been carefully 
avoided. The reader is, therefore, referred to the 
bibliography published at the end of this volume for 
matter relating to this aspect of the subject. The 
aim rather has been to outline briefly the progress of 
mathematical thought which has led up to the idea 
of the multiple dimensionality of space; to state the 

23 



24 THE MYSTERY OF SPACE 

cardinal principles of the Non-Euclidean geometry 
and to offer an interpretation of the metageometrical 
concept in the light of the evolutionary nature of 
human faculties and material characteristics and 
properties. 

The onus of this treatise is, therefore, to dis- 
tinguish between what is commonly known as sensible 
space and that other species of space known as geo- 
metric spaces. Also to show that the notion which 
has been styled hyperspace is nothing more nor less 
than an evidence of the faint, early outcroppings in 
the human mind of a faculty which, in the course of 
time, will become the normal possession of the entire 
human race. Thus the weight of all presentations will 
be to give currency to the belief, very strongly held, 
that humanity, now in its infancy, is yet to evolve 
faculties and capabilities, both mental and spiritual, 
to a degree hitherto viewed as inconceivable. 

On this view it must appear that the faculty of 
thought including the powers of imagination and con- 
ceptualization are not psychological invariants, but, on 
the other hand, are true variants. They are, conse- 
quently, answerable to the principle of evolution just 
as all vital phenomena are. Some have thought that 
no matter what idea may come into the mind of the 
human race or at what time the idea may be born 
the mind always has been able to conceive it. That 
is, many believe that the nature of mind is such that 
no matter how complex an idea may be there has 
always been in the mind the power of conceiving it. 
But this view cannot be said to have the support of 
any trustworthy testimony. If so, then the mind must 
at once be recognized as fully matured and capable 



THE PROLOGUE 25 

during every epoch of human evolution, no less in the 
first than in the latest, which, of course, is absurd. It 
is undoubtedly more reasonable and correct to believe 
that the powers of conceptualization are matters of 
evolutionary concern. For instance, the assertion that 
the mind was incapable of conceiving, in the realm 
of theology, a non-anthropomorphic god, or, in the 
field of biology, the doctrine of evolution, or, in the 
domain of invention, the wireless telegraph, or, in 
mathematics, the concept of hyperspace before the 
actual time of these conceptions, cannot be successfully 
controverted. 

In fact, it may be laid down as one of the first 
principles of psychogenesis that the mind rarely, if 
ever, conceives an idea until it has previously de- 
veloped the power of conceptualizing it and giving it 
expression in the terms of prior experience. As in 
the growth of the body there are certain processes 
which require the full development of the organ of 
expression before they can be safely executed so in 
the phyletic development of faculties there are certain 
ideas, conceptions and scopes of mental vision which 
cannot be visualized or conceptualized until the basis 
for such mentation has been laid by the appearance of 
previously developed faculties of expression. And 
especially is this true of the intellect. Inasmuch as 
the entire content of the intellect is constituted of 
sense-derived knowledge, with the exception of intui- 
tions which are not of intellectual origin though de- 
pendent upon the intellect for interpretation, there can 
be no doubt as to the necessity of there being Erst 
deposed in the intellect a sense-derived basis for in- 
tellection before it can become manifest. The Sen- 



26 THE MYSTERY OF SPACE 

sationalists, led by Leibnitz, propounded as their 
fundamental premise this dictum: "There is nothing 
in the intellect which has not first been in the senses 
except the intellect itself" and this has never been 
gainsaid by any school that could disprove it. The 
intuitionalist does not deny it: he merely claims that 
we are the recipients of another form of knowledge, 
the intuitional, which, instead of being derived from 
sense-experience, is projected into the intellectual con- 
sciousness from another source which we designate 
the Thinker. Thus, from the two forms of con- 
sciousness, come into the area of awareness truths 
that spring from entirely different sources. From the 
one source a steady stream of impressions flow con- 
stituting the substance of intellectual consciousness; 
from the other only a drop, every now and then, falls 
into the great inrushing mass so as to add a dim 
phosphorescence to an otherwise unilluminated pool. 
Obviously, when there is a lack of sensuous data from 
which a certain concept may be elaborated there can 
be no conception based upon them, and as the variety 
and quality of concepts are in exact proportion to the 
variety and quality of sense-experience there can be 
no demand for a particular species of notions such 
as might be elaborated out of the absent or non- 
existent perception. Hence, the power of conceiving 
springs forth from sense-experience. Sense-experience 
is essentially a mass of perceptions: these, creating a 
demand for additional adaptations, conspire, as if, to 
evoke the power or faculty to meet the demand, and 
consequently, an added conceptualization is made. 
Progress in human thought is made in a manner 



THE PROLOGUE 27 

similar to that which prevails in the development of 
other natural processes, such as, the power of speech 
in the child. In the development of this faculty there 
are certain definite stages which appear in due sequence. 
The child is not gifted with the power of speech at 
once. It comes, by gradual and sometimes painful 
growth, into a full use of this faculty. Now, much 
the same principle holds true in the evolution of the 
mind in the human species. It is an established 
biologic principle that the ontogenetic processes mani- 
fested in the individual are but a recapitulation of the 
phylogenetic processes which are observable in the 
progress of the entire species. The view becomes 
even more cogent when note is taken of the fact that 
the foetus, during embryogenesis, passes successively 
through stages of growth which have been shown to 
be analogous, if not identical, with those stages through 
which the human species has developed, namely, the 
mineral, vegetal and animal. 

Wherefore it may be said that the fourth dimen- 
sional concept marks a distinct stage in psychogenesis 
or evolution of mind. It required, as will be shown 
in Chapter II, nearly two thousand years for it to 
germinate, take root and come to full fruition. For 
it was not until the early years of the nineteenth cen- 
tury that mathematicians, taking inspiration from 
Riemann (1826-1866) fully recognized the concept 
as a metaphysical possibility, or even the idea was 
conceived at all. Serious doubt is entertained as to 
the possibility of its conception by any human mind 
before this date, that is, the time when it was actually 
born. Prior to that time, mathematical thought was 
taking upon itself that shape and tendence which would 



28 THE MYSTERY OF SPACE 

eventually lead to the discovery of hyperspace; but it 
could not have reached the zenith of its upward striv- 
ings at one bound. That would have been unnatural. 

Such is the constitution of the mind that although 
it is the quantity which bridges the chasm between the 
two stages of man's evolution when he merely thinks 
and when he really knows it is entirely under the 
domain of law and must observe the times and seasons, 
as it were, in the performance of its functions. The 
scope of psychogenesis is very broad, perhaps unlim- 
ited; but its various stages are very clearly defined 
notwithstanding the breadth of its scope of motility. 
And while the distance from moneron to man, or 
from feeling to thinking is vast, the gulf which sepa- 
rates man, the Thinker, from man, the knower, is 
vaster still. Who, therefore, can say what are the 
delights yet in store for the mind as it approaches, 
by slow paces, the goal whereat it will not need to 
struggle through the devious paths of perceiving, con- 
ceiving, analyzing, comparing, generalizing, inferring 
and judging; but will be able to know definitely, abso- 
lutely and instantaneously? That some such consum- 
mation as this shall crown the labors of mental evolu- 
tion seems only natural and logical. 

It may be thought by some that the character and 
content of revelational impressions constitute a varia- 
tion from the requirements of the law above referred 
to, but a little thought will expose the fallacy of this 
view. The nature of a revealed message is such as 
to make it thoroughly amenable to the restrictions 
imposed by the evolutionary aspects of mind in general. 
That this is true becomes apparent upon an ex- 
amination of the four cardinal characteristics of such 



THE PROLOGUE 29 

impressions. First, we have to consider the indefinite 
character of an apocalyptic ideograph which is due 
to its symbolic nature. This is a feature which relieves 
the impression of any pragmatic value whatsoever, 
especially for the period embracing its promulgation. 
Then, such cryptic messages may or may not be un- 
derstood by the recipient in which latter case it is 
nonpropagable. Second, the necessity of previous 
experience in the mind of the recipient in order that 
he may be able to interpret to his own mind the psychic 
impingement. The basis which such experience affords 
must necessarily be present in order that there may 
be an adequate medium of mental qualities and powers 
in which the ideogram may be preserved. A third 
characteristic is that revelations quite invariably pre- 
suppose a contemplative attitude of mind which, in 
the very nature of the case, superinduces a state of 
preparedness in the mind for the proper entertainment 
of the concept involved. This fact proves quite con- 
clusively that revelational impressions are not excep- 
tions to the general rule. Lastly, a dissatisfaction 
with the conditions with which the symbolism deals 
or to which it pertains is also a prerequisite. This 
condition is really that which calls forth the cryptic 
annunciation, and yet, preceding it is a long series of 
causes which have produced both the conditions and 
the revolt which the revelator feels at their presence. 
In view of the foregoing, it would appear that ob- 
jections based upon the alleged nonconformity of the 
revealed or inspired cannot be entertained as it must 
be manifest that it, too, falls within the scope of the 
laws of mental growth. 

Discoveries, whether of philosophical or mechanical 



3 o THE MYSTERY OF SPACE 

nature, or whether of ethical or purely mathematical 
tendence, are never the results of a deliberate, 
methodical or purposive reflection. For instance, let 
us take Lie's "transformation groups," mathematic 
contrivances used in the solution of certain theorems. 
Now, it ought to be obvious that these mathetic 
machinations were not discovered by Sophus Lie as 
a consequence of any methodic or purposeful inten- 
tion on his part. That is, he did not set out deliber- 
ately to discover "transformation groups." For back 
of the "groups" lay the entire range of analytic in- 
vestigations; the mathematical thought of more than 
a thousand years furnished the substructure upon which 
Lie built the conception of his "groups." Similarly, 
it may be said with equal assurance that no matter 
how great the intensity of thought, nor how purpose- 
ful, nor of how long duration the series of concen- 
trated abstractions which led up to the invention of 
the printing press, the linotype or multiplex printing 
press of our day could not have been produced ab- 
ruptly, nor by use of the mental dynamics of the 
human mind of remoter days. Its production had 
to follow the path outlaid by the laws of psychogenesis 
and await the development of those powers which 
alone could give it birth. The whole question re- 
solves itself, therefore, into the idea of the complete 
subserviency of the mind, in all matters of special 
moment, to the laws aforementioned. The superses- 
sion of the law of its own life by the mind is well-nigh 
unthinkable, if not quite so. 

If we now view the history of the mind as mani- 
fested in the human species, three great epochs which 
divide the scope of mental evolution into more or 



THE PROLOGUE 31 

less well-defined stages present themselves. These 
are: first, the formative stage; second, the determi- 
native stage; third, the stage of freedom, or the 
elaborative stage. 

In all of the early races of men, through every 
step which even preceded the genus homo, the generic 
mind was being formulated. It was being given shape, 
outline and direction. All of the first stage, the 
formative, was devoted to organization and direction. 
Those elementary sensations which constituted the 
basis of mind in the primitive man were accordingly 
strongly determinative of what the mind should be in 
these latter days. To this general result were con- 
tributed the effects of the activity of cells, nerves, 
bones, fibers, muscles and the blood. 

The formative period naturally covered a very 
extensive area in the history of mind or psychogenetic 
development. It was followed closely, but almost in- 
sensibly, by the determinative period during which all 
the latent powers, capacities and faculties which were 
the direct products of the formative period were being 
utilized in meeting the demands of the law of neces- 
sity. The making of provisions against domestic 
want, against the attacks of external foes; the com- 
bating of diseases, physical inefficiency, the weather, 
wild beasts, the asperities of tribal enmities; as well 
as furthering the production of art, music, sculpture, 
the various branches of handiwork, literature, philoso- 
phies, religions and the effectuation of all those 
things which now appear as the result of the mental 
activity of the present-day man make up the essence 
and purpose of the determinative period. 

Signs of the dawn of the elaborative stage, also 



32 THE MYSTERY OF SPACE 

called the stage of freedom, have been manifest now 
for upwards of three centuries and it is, therefore, 
in its beginnings. It is not fully upon us. Not yet 
can we fully realize what it may mean, nor can we 
unerringly forecast its ultimate outcome; but we feel 
that it is even now here in all the glories of its 
matutinal freshness. And the mind is beginning to be 
free from the grinding necessities of the constructive 
period having already freed itself from the restrictive 
handicaps of the primeval formulation period. AJ- 
ready the upgrowing rejuvenescences so common at 
the beginning of a new period are commencing to 
show themselves in every department of human ac- 
tivity in the almost universal desire for greater free- 
dom. And this is particularly noticeable in the many 
political upheavals which, from time to time, are 
coming to the surface as well as in the countless other 
aspects of the wide-spread renaissance. Perhaps the 
time may come, never quite fully, when there will be 
no longer any necessity to provide against the external 
exigencies of life; perhaps, the time will never be 
when the mind shall no more be bound by the law 
of self-preservation, not even when it has attained 
unto the immortality of absolute knowledge; yet, it is 
intuitively felt that it must come to pass that the mind 
shall be vastly freer than it is to-day. And with this 
new freedom must come liberation from the necessities 
of the elementary problems of mere physical existence. 
The inference is, therefore, drawn that the fourth 
dimensional concept, and all that it connotes of hyper- 
space or spaces of ^-dimensionality are some of the 
evidences that this stage of freedom is dawning. And 
the rr.ind, joyous ?.t the prospect of unbounded liberty 



THE PROLOGUE 33 

which these concepts offer, cannot restrain itself but 
has already begun to revel in the sunlit glories of 
a newer day. What the end shall be; what effect 
this new liberty will have on man's spiritual and 
economic life; and what it may mean in the upward 
strivings of the Thinker for that sublime perpetuity 
which is always the property of immediate knowledge 
no one can hope, at the present time, to fathom. It 
is, however, believed with Keyser that "it is by the 
creation of hyperspaces that the rational spirit 
secures release from limitation"; for, as he says, "in 
them it lives ever joyously, sustained by an unfailing 
sense of infinite freedom." 

The elevating influence of abstract thinking, such 
as excogitation upon problems dealing with entities 
inhabiting the domain of mathesis is, without doubt, 
incalculable in view of the fact that it is only through 
this kind of thought that the spirit is enabled to reach 
its highest possibilities. This is undoubtedly the 
philosophy of those religious and occult exercises 
known as "meditations," and this perhaps was the 
main idea in the mind of the Hebrew poet when he 
exclaimed: "Let the words of my mouth and the 
meditation of my heart be acceptable in thy sight, O 
Lord, my strength and my Redeemer." The prin- 
cipal, if not the only, value possessed by the "sum- 
mitless hierarchies of hyperspaces" which the mathe- 
matician constructs in the world of pure thought is 
the enrichening and ennobling influence which they 
exert upon the mind. But admittedly this unbounded 
domain of mathetic territory which he explores and 
which he finds "peopled with ideas, ensembles, propo- 
sitions, relations and implications in endless variety 



34 THE MYSTERY OF SPACE 

and multiplicity" is quite real to him and subsists 
under a reign of law the penalties of which, while not 
as austere and unreasonable as some which we find in 
our tridimensional world, are nevertheless quite as 
palpable and as much to be feared. For the orthodoxy 
of mathematics is as cold and intolerant as ever the 
religious fanatic can be. But the reality and even the 
actuality which may be imputed to the domain of 
mathesis is of an entirely different quality from that 
which we experience in our world of triune dimen- 
sionality and it is a regrettable error of judgment to 
identify them. It ought, therefore, never be expected, 
nor is it logically reasonable to assume that the en- 
tities which inhabit the mathetic realm of the analyst 
should be submissive to the laws of sensible space; 
nor that the conditions which may be found therein 
can ever be made conformable to the conditions which 
exist in perceptual space. 

It was Plato's belief that ideas alone possessed 
reality and what we regard as actual and real is on 
account of its ephemerality and evanescence not real 
but illusionary. This view has been shared by a number 
of eminent thinkers who followed, with some ostenta- 
tion, the lead established by Plato. For a con- 
siderable period of time this school of thinkers had 
many adherents; but the principles at length fell into 
disrepute owing to the absurdities indulged in by some 
of the less careful followers. The realism, or for 
that matter, the actuality of ideas cannot be denied; 
yet it is a realism which is neither to be compared 
with the physical reality of sense-impressions nor its 
phenomena. The character and peculiarity of ideas 
are in a class apart from similar notions of perceptual 



THE PROLOGUE 3S 

space content. It is as if we were considering the 
potentialities of the spirit world and the entities 
therein in connection with incarnate entities which in 
the very nature of the case is not allowable. Further- 
more, it is unreasonable to suppose that the conditions 
on a higher plane than the physical can be made re- 
sponsible to a similar set of conditions on the physical 
plane. 

There are certain astronomers who base their 
speculations as to the habitability of other planets 
upon the absurd hypothesis that the conditions of life 
upon all planets must be the same as those on the 
earth, forgetting that the extent of the universe and 
the scope of motility of life itself are of such a nature 
as to admit of endless variations and adaptations. 
There is a realism of ideas and a realism of per- 
ceptual space. Yet this is no reason why the two 
should be identified. On the other hand, owing to 
the diversity in the universe, every consideration would 
naturally lead to the assumption that they are dis- 
similar. To invest ideas, notions, implications and 
inferences with a reality need not logically or other- 
wise affect the reality of a stone, a fig, or even of a 
sense-impression. 

To a being on the spirit levels our grossest reali- 
ties must appear as non-existent. They are neither 
palpable nor contactable in any manner within the 
ordinary range of physical possibilities. For us his 
gravest experiences can have no reality whatsoever; for 
no matter how real an experience may be to him it is 
altogether beyond our powers of perception, and 
therefore, to us non-existent also. It should, however, 
be stated that the state of our knowledge about a 



36 THE MYSTERY OF SPACE 

given condition can in no way affect its existence. It 
merely establishes the fact that two or more realities 
may exist independent of one another and further that 
the gamut of realism in the universe is infinite and 
approaches a final state when its occlusion into abso- 
lute being follows as a logical sequence. 

Recurring to the consideration of the reality of 
spirit-realms as compared with that of sensible space, 
it comes to view that our idealism, that is, the idealism 
which is a quality of conceptualization, may be re- 
garded as identical with their realism, at least as 
being on the same plane as it. Stated differently, the 
things that are ideal to us and which constitute the 
data of our consciousness may be as real to them as 
the commonest object of sense-knowledge is to us. 
What, therefore, appears to us as the most ethereal 
and idealistic may have quite a realistic character for 
them. 

Ultimately, however, and in the final deeps of 
analysis it will be found undoubtedly that both our 
realism and our idealism as well as similar qualities 
of the spirit world are in all essential considerations 
quite illusionary. All knowledge gained in a condition 
short of divinity itself is sadly relative. Even mathe- 
matical knowledge falls far short of the absolute, the 
fondest claims of the orthodox mathematician to the 
contrary notwithstanding. It has been said frequently 
that a mathematical fact is an absolute fact and that 
its verity, necessity and certainty cannot be questioned 
anywhere in the universe whether on Jupiter, Nep- 
tune, Fomalhaut, Canopus or Spica. But having so 
declared, the fact of the sheer relativity of our 
knowledge is not disturbed thereby nor controverted. 



THE PROLOGUE 37 

Happily, neither distance nor a lack of distance can 
in any way affect the quality of human knowledge, 
mathematical knowledge not excepted. That can only 
be affected by conditions which cause it to approach 
perfection and nothing but evolution can do that. 

In the light of results obtained in analytic investi- 
gations the question of the flexibility of mathematical 
applications becomes evident and one instead of being 
convinced of the vaunted invariability of the laws 
obtaining in the world of mathesis is, on the other 
hand, made aware of the remarkable and seemingly 
unrestrained facility with which these laws may be 
made to apply to any conditions or set of assumptions 
within the range of the mind's powers of conception. 
Mathematicians have deified the definition and en- 
dowed it with omnific powers imputing unto it all the 
attributes of divinity — immutability, invariance, and 
sempiternity. In this they have erred grievously al- 
though, perhaps, necessarily. Mathetic conclusions 
are entirely conditional and depend for their certainty 
upon the imputed certitude of other propositions which 
in turn are dependent, in ever increasing and end- 
lessly complex relations, upon previously assumed 
postulates. These facts make it exceedingly difficult 
to understand the attitude of mind which has obscured 
the utter mutability and consequent ultimate unrelia- 
bility of the fine-spun theories of analytic machinations. 

The apriority of all mathematical knowledge is 
open to serious questioning. And although there is 
no hesitancy in admitting the basic agreement of the 
most primary facts of mathematical knowledge with 
the essential character of the intellect the existence 
of well-defined limits for such congruence cannot be 



38 THE MYSTERY FO SPACE 

gainsaid. The subjunctive quality of geometric and 
analytical propositions is made apparent by an ex- 
amination of the possibilities falling within the scope 
of permissibility offered by mathetic license. For in- 
stance, privileged to proceed according to the analytic 
method it is allowable to reconstruct the sequence of 
values in our ordinary system of enumeration so as 
to admit of the specification of a new value for say, 
the entire series of odd numbers. This value might 
be assumed to be a plus-or-minus one, dependent upon 
its posture in the series. That is, all odd numbers 
in the series beginning with the digit 3, and continuing, 
5, 7, 9, 11, 13, 15, 17, 19, . . . n, could be assumed 
to have only a place value which might be regarded 
as a constant-variable. The series of even numbers, 
2, 4, 6, 8, 10, 12, 14, 16, . . . n, may be assumed 
to retain their present sequence values. Under this 
system the digit 1 would have an absolute value; all 
other odd numbers would have a constant-variable 
value; constant, because always no more nor less than 
1 dependent upon their place in the operations and 
whether their values were to be applied by addition 
or subtraction to or from one of the values in the 
even number series; variable, because their values 
would be determinable by their application and alge- 
braic use. 

There would, of course, be utilitarian objection to 
a system of this kind; but under the conditions of a 
suppositionary hypothesis, it would be self-consistent 
throughout, and if given universal assent would suit 
our purposes equally as w T ell as our present system. 
But the fact that this can be done under the mathe- 
matic method verily proves the violability of mathe- 



THE PROLOGUE 39 

matical laws and completely negatives the assumption 
that the sum of any two digits, as say 2 plus 2 equals 
4, is necessarily and unavoidably immutable. For 
it can be seen that the sum-value of all numbers may 
be made dependent upon the assumed value which may 
be assigned to them or to any collection thereof. 
Furthermore, it is a matter of historical knowledge 
that it was the custom of ancient races of men to 
account for values by an entirely different method from 
what we use to-day. The latter is a result of evolu- 
tion and while experience teaches that it is by far the 
most convenient, it is nevertheless true that earlier 
men managed at least fairly well on a different basis. 
Then, too, the fact of the utility and universal ap- 
plicability of our present system, based upon universal 
assent, does not obviate the conclusion that any other 
system, consistent in itself, might be made to serve our 
purposes as well. 

It ought to be said, however, in justice to the rather 
utilitarian results obtained by La Grange, Helm- 
holtz, Fechner, and others who strove to make use 
of their discoveries in analysis in solving mechanical, 
physiological and other problems of more or less 
pragmatic import that, in so far as this is true, mathe- 
matical knowledge must be recognized as being con- 
sistent with the necessities of a priori requirements. 
But even these results may not be regarded as tran- 
scending the scope of the most fundamental principles 
of sense-experience. It will be discovered finally, 
perhaps, that the energy spent in elaborating compli- 
cate series of analytic curiosities has been misappro- 
priated. It will then be necessary to turn the attention 
definitely to the study of that which lies not at the 



4 o THE MYSTERY OF SPACE 

terminus of the intellect's modus vivendi, but which 
is both the origin of the intellect and its eternal sus- 
tainer — the intuition, or life itself. This can result 
in nothing less than the complete spiritualization of 
man's mental outlook and the consequent inevitable 
recognition of the underlying and ever-sustaining 
one-ness of all vital manifestations. 

One of the curiosities of the tendency in man's 
mind to specialize in analytics, whether in the field of 
pure mathematics or metaphysics, is the fact that it 
almost invariably leads to an attempt to account for 
cosmic origins on the basis of paralogic theories. This 
in times past has given rise to the theory of the purely 
mechanical origin of the universe as well as many 
other fantastic fallacies the chief error of which lay 
in the failure to distinguish between the realism of 
mental concepts and that of the sensible world. In 
spite of this, however, one is bound to appreciate the 
beneficial effects of analytic operations because they 
serve as invigorants to mental growth. It could not, 
therefore, be wished that there were no such thing as 
analytics; for the equilibria-restoring property of the 
mind may at all times be relied upon to minimize the 
danger of excesses in either direction. Just as the 
tide flowing in flows out again, thereby restoring the 
ocean's equilibrium, so the mind ascending in one 
generation beyond the safety mark has its equilibrium 
restored in the next by a relinquishment of the follies 
of the former. 

The four-space is one of the curiosities of analytics; 
yet it need not be a menace to the sane contempla- 
tion of the variegated products of analysis. Safety 
here abides in the restraint which should characterize 



THE PROLOGUE 41 

all discussion and application of the concept. If en- 
thusiasts would be content not to transport the so- 
called fourth dimensional space out of the sphere of 
hyperspace and cease trying to speculate upon the 
results of its interposal into three space conditions, 
which is in every way a constructual impossibility, 
there could not be any possible objection to its due 
consideration. This would obviate the danger of 
calling into question either the sincerity or perspicacity 
of those whose enthusiasm tempts them to transgress 
the limits of propriety in their behavior towards the 
inquiry. 

There is but one life, one mind, one extension, 
one quantity, one quality, one being, one state, one 
condition, one mood, one affection, one desire, one 
feeling, one consciousness. There is also but one 
number and that is unity. All so-called integers are 
but fractional parts of this kosmic unity. The idea 
represented by the word two really connotates two 
parts of unity and the same is true of a decillion, or 
any number of parts. These are merely the infi- 
nitesimals of unity and they grow less in size and 
consequence as the divisions increase in number. The 
analysis of unity into an infinity of parts is purely an 
a posteriori procedure. That it is an inherent mind- 
process is a fallacy. All our common quantities, as 
the mile, kilometer, yard, foot, inch, gallon, quart, are 
conventional and arbitrary and susceptible of wide 
variations. As the basis of all physical phenomena 
is unity; it is only in the ephemeral manifestations of 
sensuous objects that they appear as separate and 
distinct quantities. 

We see on a tree many leaves, many apples or 



42 THE MYSTERY OF SPACE 

cherries; on a cob many grains of corn. We have 
learned to assign to each of these quantities in their 
summation a sequence value. But this is an empirical 
notion and cannot be said to inhere in the mind itself. 
Let us take, for instance, the mustard seed. If it 
were true that in one of these seeds there existed all 
the subsequent seeds which appear in the mustard 
plant as separate and identifiable quantities, and not 
in essence, then there would perhaps be warrant for 
the notion that diversity, as the calculable element, is 
an a priori conception. But, as this is not the case and 
since diversity is purely empirical and pertains only to 
the efflorescence of the one life it is manifestly absurd 
to take that view. 

Under the most charitable allowances, therefore, 
there can be but two quantities — unity and diversity; 
yet not two, for these are one. Unity is the one 
quantity and diversity is the division of unity into a 
transfinity of parts. Unity is infinite, absolute and 
all-inclusive. Diversity is finite although it may be 
admitted to be transfinite, or greater than any assign- 
able value. Unity alone is incomprehensible. In order 
to understand something of its nature we divide it into 
a diversity of parts; and because we fail to understand 
the transfinity of the multitude of parts we mistakenly 
call them infinite. 

When analysis shall have proceeded far enough 
into the abysmal mysteries of diversity; when the 
mathematical mind shall have been overcome by the 
overwhelming perplexity of the maze of diverse parts, 
it shall then fall asleep and upon awaking shall find 
that wonderfully simple thing — unity. It is the one 
quantity that is endowed with a magnitude which is 



THE PROLOGUE 43 

both inconceivable and irresolvable. The one in- 
eluctable fact in the universe is the incomprehensibility 
and all-inclusivity of one-ness. It is incomprehensible, 
inconceivable and infinite at the present stage of mind 
development. But the goal of mind is to understand 
the essential character of unity, of life. Its evolution 
will then stop, for it will have reached the prize of 
divinity itself whereupon the intellect exalted by and 
united with the intuition shall also become one with 
the divine consciousness. 



CHAPTER II 

Historical Sketch of the Hyperspace 
Movement 

Egypt the Birthplace of Geometry — Precursors: Nasir-Eddin, 
Christoph Clavius, Saccheri, Lambert, La Grange, 
Kant — Influence of the Mecanique Analytique — The 
Parallel-Postulate the Root and Substance of the Non- 
Euclidean Geometry — The Three Great Periods: The 
Formative, Determinative and Elaborative — Riemann and 
the Properties of Analytic Spaces. 

The evolution of the idea of a fourth dimension 
of space covers a long period of years. The earliest 
known record of the beginnings of the study of space 
is found in a hieratic papyrus which forms a part of 
the Rhind Collection in the British Museum and which 
has been deciphered by Eisenlohr. It is believed 
to be a copy of an older manuscript of date 3400 B. C, 
and is entitled "Directions for Knowing All Dark 
Things!* The copy is said to have been made by 
Ahmes, an Egyptian priest beween 1700 and 11 00 
B. C. It begins by giving the dimensions of barns; 
then follows the consideration of various rectilineal 
figures, circles, pyramids, and the value of pi (71-). 
Although many of the solutions given in the manu- 
script have been found to be incorrect in minor par- 
ticulars, the fact remains that Egypt is really the 
birth-place of geometry. And this fact is buttressed 

44 



SKETCH OF HYPERSPACE MOVEMENT 45 

by the knowledge that Thales, long before he 
founded the Ionian School which was the beginning 
of Greek influence in the study of mathematics, is 
found studying geometry and astronomy in Egypt. 

The concept of hyperspace began to germinate in 
the latter part of the first century, B. C. For it was 
at this date that Geminos of Rhodes (b. c. 70) be- 
gan to think seriously of the mathematical labyrinth 
into which Euclid's parallel-postulate most certainly 
would lead if an attempt at demonstrating its certitude 
were made. He recognized the difficulties which 
would engage the attention of those who might venture 
to delve into the mysterious possibilities of the prob- 
lem. There is no doubt, too, but that Euclid him- 
self was aware, in some measure at least, of these 
difficulties; for his own attitude towards this postulate 
seems to have been one of noncommittance. It is, 
therefore, not strange that the astronomer, Ptolemy 
(a. d. 87-165), should be found seeking to prove the 
postulate by a consideration of the possibilities of 
interstellar triangles. His researches, however, brought 
him no relief from the general dissatisfaction which 
he felt with respect to the validity of the problem 
itself. 

For nearly one thousand years after the attempts 
at solving the postulate by Geminos and Ptolemy, the 
field of mathematics lay undisturbed. For it was at 
this time that there arose a strange phenomenon, 
more commonly known as the "Dark Ages," which 
put an effectual check to further research or inde- 
pendent investigations. Mathematicians throughout 
this long lapse of time were content to accept Euclid 
as the one incontrovertible, unimpeachable authority, 



46 THE MYSTERY OF SPACE 

and even such investigations as were made did not 
have a rebellious tendence, but were mainly endeavors 
to substantiate his claims. 

Accordingly, it was not until about the first half 
of the thirteenth century that any real advance was 
made. At this time there appeared an Arab, Nasir- 
Eddin ( i 201-1274) who attempted to make an im- 
provement on the problem of parallelism. His work 
on Euclid was printed in Rome in 1594 A. D., about 
three hundred and twenty years after his demise and 
was communicated in 1651 by John Wallis (1616- 
1703) to the mathematicians of Oxford University. 
Although his calculations and conclusions were re- 
spectfully received by the Oxford authorities no 
definite results were regarded as accomplished by 
what he had done. It is believed, however, that his 
Work reopened speculation upon the problem and 
served as a basis, however slight, for the greater work 
that was to be done by those who followed him during 
the next succeeding eight hundred years. 

About twenty years before the printing of the 
work of Nasir-Eddin, Christoph Clavius (1574) 
deduced the axiom of parallels from the assumption 
that a line whose points are all equidistant from a 
straight line is itself straight. In his consideration of 
the parallel-postulate he is said to have regarded it 
as Euclid's Xlllth axiom. Later Bolyai spoke of 
it as the Xlth and later still, Todhunter treated it 
as the Xllth. Hence, there does not seem to have 
been any general unanimity of opinion as to the exact 
status of the parallel-postulate, and especially is this 
true in view of the uncertainty now known to have 
existed in Euclid's mind concerning it. 



SKETCH OF HYPERSPACE MOVEMENT 47 



Girolamo Saccheri (1667-1733), a learned 
Jesuit, born at San Remo, came next upon the stage. 
And so important was his work that it will perpetuate 
the memory of his name in the history of mathematics. 
He was a teacher of grammar in the Jesuit Collegio 
di Brera where Tommaso Ceva, a brother of Gio- 
vanni, the well-known mathematician, was teacher of 
mathematics. His association with the Ceva brothers 
was especially beneficial to him. He made use of 
Ceva's very ingenious methods in his first published 
book, 1693, entitled Solutions of Six Geometrical 
Problems Proposed by Count Roger Ventimiglia. 



Fig. 1. 

Saccheri attacked the problem of parallels in 
quite a new way. Examining a quadrilateral, ABCD, 
in which the angles A and B are right angles and the 
sides AC and BD are equal, he determined to show 
that the angles C and D are equal. He also sought 
to prove that they are either right angles, obtuse or 
acute. He undertook to prove the falsity of the latter 
two propositions (that they are either obtuse or acute), 
leaving as the only possibility that they must be right 
angles. In doing so, he found that his assumptions 
led him into contraditions which he experienced diffi- 
culty in explaining. 

His labors in connection with the solution of the 
problems proposed by Count Ventimiglia, includ- 



4 S THE MYSTERY OF SPACE 

ing his work on the question of parallels, led directly 
into the field of metageometrical researches, and per- 
haps to him as to no other who had preceded him, 
or at least to him in a larger degree, belongs the credit 
for a continued renewal of interest in that series of 
investigations which resulted in the formulation of the 
non-Euclidean geometry. 

The last published work of Saccheri was a recital 
of his endeavors at demonstrating the parallel-postu- 
late. This received the "Imprimatur" of the In- 
quisition, July 13, 1733; the Provincial Company of 
Jesus took possession of the book for perusal on 
August 16,1733 ; but unfortunately within two months 
after it had been reviewed by these authorities, 
Saccheri passed away. 

All efforts which had been made prior to the work 
of Saccheri were based upon the assumption that 
there must be an equivalent postulate which, if it could 
be demonstrated, would lead to a direct, positive proof 
of Euclid's proposition. Although these and all 
other attempts at reaching such a proof have signally 
failed and although it may correctly be said that the 
entire history of demonstrations aiming at the solu- 
tion of the famous postulate has been one long series 
of utter failures, it can be asserted with equal certitude 
that it has proven to be one of the most fruitful prob- 
lems in the history of mathematical thought. For out 
of these failures has been built a superstructure of 
analytical investigations which surpasses the most 
sanguine expectations of those who had labored and 
failed. 

In 1766 John Lambert (1728-1777) wrote a 
paper upon the Theory of Parallels dated Sept. 



SKETCH OF HYPERSPACE MOVEMENT 49 

5, 1766, first published in 1786, from the papers left 
by F. Bernoulli, which contained the following 
assertions i 1 

1. The parallel-axiom needs proof, since it does 
not hold for geometry on the surface of the sphere. 

2. In order to make intuitive a geometry in which 
the triangle's sum is less than two right angles, we 
need an "imaginary" sphere (the pseudosphere). 

3. In a space in which the triangle's sum is differ- 
ent from two right angles there is an absolute measure 
(a natural unit for length). 

At this time Immanuel Kant (1 724-1 804), the 
noted German metaphysician, was in the midst of his 
philosophical labors. And it is believed that it was 
he who first suggested the idea of different spaces. 
Below is given a statement taken from his Prolego- 
mena 2 which corroborates this view. 

"That complete space (which is itself no longer 
the boundary of another space) has three dimen- 
sions, and that space in general cannot have more, 
is based on the proposition that not more than three 
lines can intersect at right angles in one point. . . . 
That we can require a line to be drawn to infinity, 
a series of changes to be continued (for example, 
spaces passed through by motion) in indefinitum, 
presupposes a representation of space and time 
which can only attach to intuition." 

His differentiation between space in general and 
space which may be considered as the "boundary of 
another space" shows, in the light of the subsequent 

1 Vide New York Mathematical Society Bulletin, Vol. Ill, 1893-4, 
p. 79, G. B. Halstead on Lambert's Non-Euclidean Geometry. 

2 Prolegomena, Kant, p. 37, Trans, by J. P. Mahaffy and J. H. 
Bernard, 



50 THE MYSTERY OF SPACE 

developments of the mathematical idea of space that 
he very fully appreciated the marvelous scope of 
analytic spaces. His conception of space, therefore, 
must have had a profound influence upon the mathe- 
matic thought of the day causing it to undergo a rapid 
reconstruction at the hands of geometers who came 
after him. 

Under the masterly influence of La Grange 
(1736-18 13) the idea of different spaces began to take 
definite shape and direction; the geometry of hyper- 
space began to crystallize; and the field of mathesis 
prepared for the growth of a conception the com- 
prehension of which was destined to be the pro- 
foundest undertaking ever attempted by the human 
mind. Unlike most great men whom the world learns 
tardily to admire, La Grange lived to see his talents 
and genius fully recognized by his compeers; for he 
was the recipient of many honors both from his 
countrymen and his admirers in foreign lands. He 
spent twenty years in Prussia where he went upon the 
invitation of Frederick the Great who in the Royal 
summons referred to himself as the "greatest king 
in Europe" and to La Grange as the "greatest 
mathematician" in Europe. In Prussia the Mecanique 
Analytique and a long series of memoirs which were 
published in the Berlin and Turin Transactions were 
produced. La Grange did not exhibit any marked 
taste for mathematics until he was 17 years of age. 
Soon thereafter he came into possession of a memoir 
by Halley quite by accident and this so aroused his 
latent genius that within one year after he had re- 
viewed Halley's memoir he became an accomplished 
mathematician. 



SKETCH OF HYPERSPACE MOVEMENT 51 

He created the calculus of variations, solved most 
of the problems proposed by Fermat, adding a num- 
ber of theorems of his own contrivance; raised the 
theory of differential equations to the position of a 
science rather than a series of ingenious methods for 
the solution of special problems and furnished a solu- 
tion for the famous isoperimetrical problem which 
had baffled the skill of the foremost mathematicians 
for nearly half a century. All these stupendous tasks 
he performed by the time he reached the age of 
nineteen. 

The Mecanique Analytique is his greatest and 
most comprehensive work. In this he established the 
law of virtual work from which, by the aid of his 
calculus of variations, he deduced the whole of 
mechanics, including both solids and liquids. It was 
his object in the Analytique to show that the whole sub- 
ject of mechanics is implicitly embraced in a single 
principle, and to lay down certain formulae from which 
any particular result can be obtained. He frequently 
made the assertion that he had, in the Mecanique 
Analytique, transformed mechanics which he per- 
sistently defined as a "geometry of four dimensions" 1 
into a branch of analytics and had shown the so-called 
mechanical principles to be the simple results of the 
•calculus. Hence, there can be no doubt but that La 

"In 1754 D'Alembert (1717-1783) published an article in 
the famous old Encyclopedia edited by Diderot and himself on 
Dimension. In this article the idea of the fourth dimension is 
dwelt upon at length. The view which he expressed in this article, 
of course, served greatly to popularize the conception among the 
learned men of the day, and owing to the close relationship exist- 
ing between D'Alembert and La Grange, it is not surprising that 
the latter should have been very much enamored of the idea. 



S 2 THE MYSTERY OF SPACE 

Grange not only completed the foundation, but pro- 
vided most of the material in his analyses and other 
"abstract results of great generality" which he ob- 
tained in his numerous calculations, for the super- 
structure subsequently known as the geometry of 
hyperspace, and in which the fourth dimensional con- 
cept occupies a very fundamental place. 

It is as if for nearly seventeen hundred years work- 
men, such as Geminos, of Rhodes, Ptolemy, Sac- 
cheri, Nasir-Eddin, Lambert, Clavius, and hun- 
dred of others who struggled with the problem of 
parallels, had made more or less sporadic attempts at 
the excavation of the land whereon a marvelously 
intricate building was to be constructed. There is no 
historical evidence to show that any of them ever 
dreamed that the results of their labors would be 
utilized in the manner in which they have been used. 
Then came Kant with the wonderfully penetrating 
searchlight of his masterful intellect who from the 
elevation which he occupied saw that the site had great 
possibilities, but he had not the mathematical talent 
to undertake the work of actual, methodical construc- 
tion. Indeed his task was of a different sort. How- 
ever, he succeeded in opening the way for La Grange 
and others who followed him. La Grange immedi- 
ately seized upon the idea which for more than a 
thousand years had been impinging upon the minds 
of mathematicians vainly seeking lodgment and began 
the elaboration of a plan in accordance with which 
minds better skilled in the pragmatic application of 
abstract principles than his could complete the work 
begun. Unfortunately, on account of his intense de- 
votion and loyalty to the study of pure mathematics, 



SKETCH OF HYPERSPACE MOVEMENT 53 

and when he had reached the summit of his greatness 
where he stood "without a rival as the foremost living 
mathematician," his health became seriously affected, 
causing him to suffer constant attacks of profound 
melancholia from which he died on April 10, 18 13. 

We come now to one of the most remarkable 
periods in the history of mental development. Dur- 
ing the six hundred years between the birth of Nasir- 
Eddin and the death of La Grange the entire world 
of mathesis was being reconstituted. Since there had 
been gradually going on an internal process which, 
when completed, forever would liberate the mind 
from the narrow confines of consciousness limited to 
the three-space, it is not surprising that we should find, 
in the mathematical thought of the time, an absolutely 
epoch-making departure. The innumerable attempts 
at the solution of the parallel-postulate, all failures in 
the sense that they did not prove, have intensified 
greatly the esteem in which the never-dying elements 
of Euclid are held to-day. And despite the fact that 
there may come a time when his axioms and conclu- 
sions may be found to be incongruent with the facts 
of sensuous reality; and though all of his fundamental 
conceptions of space in general, his theorems, proposi- 
tions and postulates may have to give way before the 
searching glare of a deeper knowledge because of 
some revealed fault, the perfection of his work in 
the realm of pure mathematics will remain forever a 
master piece demanding the undiminished admiration 
of mankind. 

The parallel-postulate, as stated by EUCLID in his 
Elements of Geometry, reads as follows: 



54 THE MYSTERY OF SPACE 

"If a straight line meet two straight lines so as 
to make the two interior angles on the same side 
of it taken together less than two right angles, 
these straight lines being continually produced, shall 
at length meet upon that side on which are the 
angles which are less than two right angles." 

On this postulate hang all the u law and the 
prophets" of the non-Euclidean Geometry. In it 
are the virtual elements of three possible geometries. 
Furthermore, it is both the warp and the woof of 
the loom of present-day metageometrical researches. 
It is the golden egg laid by the god Seb at the be- 
ginning of a new life cycle in psychogenesis. Its 
progeny are numerous — hyperspaces, sects, straights, 
digons, equidistantials, polars, planars, coplanars, in- 
variants, quaternions, complex variables, groups and 
many others. A wonderfully interesting breed, full 
of meaning and pregnant with the power of final 
emancipations for the human intellect! 

When the conclusions which were systematically 
formulated as a result of the investigations along the 
lines of hypotheses which controverted the parallel- 
postulate were examined it was found that they fell 
into three main divisions, namely: the synthetic or 
hyperbolic; the analytic or Riemannian and the 
elliptic or Cayley-Klein. These divisions or groups 
are based upon the three possibilities which inhere in 
the conception taken of the sum of the angles referred 
to in the above postulate as to whether it is equal to, 
greater or less than two right angles. 

The assumption that the angular sum is congruent 
to a straight angle is called the Euclidean or para- 
bolic hypothesis and is to be distinguished from the 



SKETCH OF HYPERSPACE MOVEMENT 55 

synthetic or hyperbolic hypothesis established by 
Gauss, Lobachevski and Bolyai and which assumes 
that the angular sum is less than a straight angle. 
The elliptic or Cayley-Klein hypothesis assumes that 
the angular sum is greater than a straight angle. 
Lobachevski, however, not satisfied with the state- 
ment of the parallel-postulate as given by Euclid and 
which had caused the age-long controversy, substituted 
for it the following: 

"All straight lines which, in a plane, radiate 
from a given point, can, with respect to any other 
straight line, in the same plane, be divided into two 
classes — the intersecting and the non-intersecting. 
The boundary line of the one and the other class 
is called parallel to the given line." 

This is but another way of saying about the same 
thing that Euclid had declared before, and yet, curi- 
ously enough it afforded just the liberty that Loba- 
chevski needed to enable him to elaborate his theory. 

For the purposes of this sketch the field of the 
development of non-Euclidean geometry is divided 
into three periods to be known as : ( 1 ) the formative 
period in which mathematical thought was being 
formulated for the new departure; (2) the determina- 
tive period during which the mathematical ideas were 
given direction, purpose and a general tendence; (3) 
the elaborative period during which the results of the 
former periods were elaborated into definite kinds of 
geometries and attempts made at popularizing the 
hypotheses. 



56 THE MYSTERY OF SPACE 



The Formative Period 

Charles Frederich Gauss (1777-1855) by 
some has been regarded as the most influential mathe- 
matician that figured in the formulation of the non- 
Euclidean geometry; but closer examination into his 
efforts at investigating the properties of a triangle 
shows that while his researches led to the establish- 
ment of the theorem that a regular polygon of seven- 
teen sides (or of any number which is prime, and 
also one more than a power of two) can be inscribed, 
under the Euclidean restrictions as to means, in a 
circle, and also that the common spherical angle on 
the surface of a sphere is closely connected with the 
constitution of the area inclosed thereby, he cannot 
justly be designated as the leader of those who formu- 
lated the synthetic school. And this, too, for the 
simple reason that, as he himself admits in one of 
his letters to Taurinus, he had not "published any- 
thing on the subject." In this same letter he informs 
Taurinus that he had pondered the subject for more 
than thirty years and expressed the belief that there 
could not be any one who had "concerned himself more 
exhaustively with this second part (that the sum of 
the angles of a triangle cannot be more than 1 8.0 de- 
grees)" than he had. 

Writing from Gottingen to Taurinus, November 
8, 1824, and commenting upon the geometric value of 
the sum of the angles of a triangle, he-says: 

"Your presentation of the demonstration that 
the sum of the angles of a plane triangle cannot be 
greater than 180 degrees does, indeed, leave some- 



SKETCH OF HYPERSPACE MOVEMENT 57 

thing to be desired in point of geometrical precision. 
But this could be supplied, and there is no doubt 
that the impossibility in question admits of the most 
rigorous demonstration. But the case is quite dif- 
ferent with the second part, namely, that the sum 
of the angles cannot be smaller than 180 degrees; 
this is the real difficulty, the rock upon which all 
endeavors are wrecked. . . . The assumption that 
the sum of the three angles is smaller than 180 
degrees leads to a new geometry entirely different 
from our Euclidean — a geometry which is through- 
out consistent with itself, and which I have elabo- 
rated in a manner entirely satisfactory to myself, 
so that I can solve every problem in it with the ex- 
ception of the determining of a constant which is 
not a priori obtainable." 

It appears from this correspondence that Gauss 
had in the privacy of his own study elaborated a com- 
plete non-Euclidean geometry, and had so thoroughly 
familiarized himself with its characteristics and possi- 
bilities that the solution of every problem embraced 
within it was very clear to him except that of the 
determination of a constant. He concluded the above 
letter by saying: 

U A11 my endeavors to discover contradiction or 
inconsistencies in this non-Euclidean geometry have 
been in vain, and the only thing in it that conflicts 
with our reason is the fact that if it were true there 
would necessarily exist in space a linear magnitude 
quite determinate in itself; yet unknown to us." 

Judging from the correspondence between Gauss 
and Gerling (1788-1857), Bessel (1784-1846), 
Schumacher and Taurinus, the nephew of Schwei- 



58 THE MYSTERY OF SPACE 

kart, and that between Schweikart and Gerling, 
there had grown up a general dissatisfaction in the 
minds of mathematicians of this period with Euclidean 
geometry and especially the parallel-postulate and its 
connotations. Bessel expresses this general discon- 
tent in one of his letters to Gauss, dated February 10, 
1829, in which he says: 

"Through that which Lambert said and what 
Schweikart disclosed orally, it has become clear 
to me that our geometry is incomplete, and should 
receive a correction, which is hypothetical, and if 
the sum of the angles of the plane triangle is equal 
to 180 degrees, vanishes.'' 

The opinion of leading mathematicians at this time 
seems to have been crystallizing very rapidly. Un- 
consciously the men of this formative period were 
adducing evidence which would give form and tendence 
to the developments in the field of mathesis at a later 
date. They appear to have been reaching out for 
that which, ignis fatuus-like, was always within easy 
reach, but not quite apprehensible. 

A bolder student than Gauss was Ferdinand 
Carl Schweikart (1 780-1 857) who also has been 
credited with the founding of the non-Euclidean 
geometry. In fact, if judged by the same standards 
as Gauss, he would be called the "father of the 
geometry of hyperspace"; for he really published the 
first treatise on the subject. This was in the nature 
of an inclosure which he inserted between the leaves 
of a book he loaned to Gerling. He also asked that 
it be shown to Gauss that he might give his judgment 
as to its merits. 



SKETCH OF HYPERSPACE MOVEMENT 59 

Schweikart's treatise, dated Marburg, Decem- 
ber, 1818, is here quoted in full: 

"There is a two-fold geometry — a geometry in 
the narrower sense, the Euclidean, and an astral 
science of magnitude. 

"The triangles of the latter have the peculiarity 
that the sum of the three angles is not equal to two 
right angles. 

"This presumed, it can be most rigorously 
proven: (a) That the sum of the three angles in 
the triangle is less than two right angles. 

"(&) That this sum becomes ever smaller, the 
more content the angle incloses, (c) That the alti- 
tude of an isosceles right-angled triangle indeed ever 
increases, the more one lengthens the side; that it, 
however, cannot surpass a certain line which I call 
the constant." 

Squares have consequently the following form: 




"If this constant were for us the radius of the 
earth (so that every line drawn in the universe, from 



60 THE MYSTERY OF SPACE 

one fixed star to another, distant 90 from the first, 
would be a tangent to the surface of the earth) it 
would be infinitely great in comparison with the 
spaces which occur in daily life." 

The above, being the first published, not printed, 
treatise on the new geometry occupies a unique place 
in the history of higher mathematics. It gave addi- 
tional strength to the formative tendencies which 
characterized this period and marked Schweikart as 
a constructive and original thinker. 

The nascent aspects of this stage received a fruit- 
ful contribution when Nicolai Lobachevski (1793- 
1847) created his Imaginary Geometry and Janos 
Bolyai ( 1 802-1 860) published as an appendix to his 
father's Tentamen, his Science Absolute of Space. 
Lobachevski and Bolyai have been called the 
"Creators of the Non-Euclidean Geometry." And this 
appellation seems richly to be deserved by these 
pioneers. Their work gave just the impetus most 
needed to fix the status of the new line of researches 
which led to such remarkable discoveries in the more 
recent years. The Imaginary Geometry and the 
Science Absolute of Space were translated by the 
French mathematician, J. Houel in 1868 and by him 
elevated out of their forty-five years of obscurity and 
non-effectiveness to a position where they became avail- 
able for the mathematical public. To Bolyai and 
Lobachevski, consequently, belong the honor of 
starting the movement which resulted in the develop- 
ment of metageometry and hence that which has 
proved to be the gateway of a new mathematical free- 
dom. 



SKETCH OF HYPERSPACE MOVEMENT 61 

Gauss, Schweikart, Lobachevski, Wolfgang 
and Janos Bolyai were the principal figures of the 
formative period and the value of their work with 
respect to the formulation of principles upon which 
was constructed the Temple of Metageometry cannot 
be overestimated. 

The Determinative Period 

This period is characterized chiefly by its close 
relationship to the theory of surfaces. Riemann's 
Habilitation Lecture on The Hypotheses Which Con- 
stitute the Bases of Geometry marks the beginning of 
this epoch. In this dissertation, Riemann not only 
promulgated the system upon which Gauss had spent 
more than thirty years of his life in elaborating, for 
he was a disciple of Gauss; but he disclosed his own 
views with respect to space which he regarded as a 
particular case of manifold. His work contains two 
fundamental concepts, namely, the manifold and the 
measure of curvature of a continuous manifold, pos- 
sessed of what he called flatness in the smallest parts. 
The conception of the measure of curvature is ex- 
tended by Riemann from surfaces to spaces and a 
new kind of space, finite, but unbounded, is shown 
to be possible. He showed that the dimensions of 
any space are determined by the number of measure- 
ments necessary to establish the position of a point in 
that space. Conceiving, therefore, that space is a mani- 
fold of finite, but unbounded, extension, he established 
the fact that the passage from one element of a mani- 
fold to another may be either discrete or continuous and 
that the manifold is discrete or continuous according 



62 THE MYSTERY OF SPACE 

to the manner of passage. Where the manifold is 
regarded as discrete two portions of it can be com- 
pared, as to magnitude, by counting; where continuous, 
by measurement. If the whole manifold be caused to 
pass over into another manifold each of its elements 
passing through a one-dimensional manifold, a two- 
dimensional manifold is thus generated. In this way, 
a manifold of w-dimensions can be generated. On the 
other hand, a manifold of w-dimensions can be an- 
alyzed into one of one dimension and one of (n-i) 
dimensions. 

To Riemann, then, is due the credit for first 
promulgating the idea that space being a special case 
of manifold is generable, and therefore, finite. He 
laid the foundation for the establishment of a special 
kind of geometry known as the "elliptic." Space, as 
viewed by him, possessed the following properties, 
viz. : generability, divisibility, measurability, ponder- 
ability, finity and flexity. 

These are the six pillars upon which rests the 
structure of hyperspace analyses. 3 

Generability is that property of geometric space 
by virtue of which it may be generated, or con- 
structed, by the movement of a line, plane, surface 
or solid in a direction without itself. Divisibility is 
that property of geometric space by virtue of which 
it may be segmented or divided into separate parts 
and superposed, or inserted, upon or between each 
other. Measurability is that property by virtue of 

'Vide Nature, Vol. VIII, pp. 14-17; 36, 37 (1873); also Mathe- 
matical Papers, pp. 65-71. 



SKETCH OF HYPERSPACE MOVEMENT 63 

which geometric space is determined to be a manifold 
of either a positive or negative curvature, also by 
which its extent may be measured. Ponderability is 
that property of geometric space by virtue of which it 
may be regarded as a quantity which can be manipu- 
lated, assorted, shelved or otherwise disposed of. 
Finity is that property by virtue of which geometric 
space is limited to the scope of the individual con- 
sciousness of a unodim, a duodim or a tridim and by 
virtue of which it is finite in extent. Flexity is that 
property by virtue of which geometric space is re- 
garded as possessing curvature, and in consequence of 
which progress through it is made in a curved, rather 
than a geodetic line, also by virtue of which it may be 
flexed without disruption or dilatation. 

Riemann who thus prepared the way for entrance 
into a veritable labyrinth of hyperspaces is, therefore, 
correctly styled "The father of metageometry," and 
the fourth dimension is his eldest born. He died while 
but forty years of age and never lived long enough 
fully to elaborate his theory with respect to its ap- 
plication to the measure of curvature of space. This 
was left for his very energetic disciple, Eugenio 
Beltrami (1 835-1 900) who was born nine years 
after Riemann and lived thirty-four years longer 
than he. His labors mark the characteristic stand- 
point of the determinative period. Beltrami's mathe- 
matical investigations were devoted mainly to the 
non-Euclidean geometry. These led him to the rather 
remarkable conclusion that the propositions embodied 
therein relate to figures lying upon surfaces of con- 
stant negative curvature. 



6 4 



THE MYSTERY OF SPACE 



Beltrami sought to show that such surfaces par- 
take of the nature of the pseudosphere, and in doing 
so, made use of the following illustration: 





B 

Fig. 3. 




If the plane figure aabb is made to revolve upon 
its axis of symmetry AB the two arcs, ab and ab will 
describe a pseudospherical concave^convex surface like 



SKETCH OF HYPERSPACE MOVEMENT 65 

that of a solid anchor ring. Above and below, to- 
ward aa and bb t the surface will turn outward with 
ever-increasing flexure till it becomes perpendicular to 
the axis and ends at the edge with one curvature in- 
finite. Or, the half of a pseudospherical surface may 
be rolled up into the shape of a champagne glass, as 
in Fig. 4. In this way, the two straightest lines of 
the pseudospherical surface may be indefinitely pro- 
duced, giving a kind of space (pseudospherical) in 
which the axiom of parallels does not hold true. 

The determinative period marks the most im- 
portant stage in the development of non-Euclidean 
geometry and certainly the most significant in the evo- 
lution of the idea of hyperspaces and multiple dimen- 
sionality. Riemann and Beltrami are chief among 
those whose labors characterize the scope of this 
period. Their work gave direction and general out- 
line for later developments and all subsequent re- 
searches along these lines have be'en conducted in strict 
conformity with the principles laid down by these 
pioneer constructionists. They laid out the field and 
designated its confines beyond which no adventurer has 
since dared to pass. 

The great importance of the work of Riemann at 
this time may be seen further in the fact that it not 
only marked the beginning of a new epoch in geometry; 
but his pronouncement of the hypothesis that space is 
unbounded, though finite, is really the first time in 
the history of human thought that expression was ever 
given to the idea that space may yet be only of limited 
extent. Before that time the minds of all men seemed 
to have been unanimous in the consideration of space 
as an illimitable and infinite quantity. 



66 THE MYSTERY OF SPACE 



The Elaborative Period 

The elaborative stage includes the work of all 
those who, working upon the bases laid down by 

LOBACHEVSKI, BOLYAI, SCHWEIKART and RlEMANN, 

have sought to amplify the conclusions reached by 
them. Among those whose investigations have greatly 
multiplied the applications of hyperspace conceptions 
are Houel (1866) and Flye St. Marie (1871) of 
France; Helmholtz (1868), Frischauf (1872), 
Klein (1849), an d Baltzer (1877) of Germany; 
Beltrami (1872) of Italy; De Tilly (1879) of Bel- 
gium; Clifford and Cayley (1821) of England; 
Newcomb (1835) and Halstead of America. 

These have been most active in popularizing the 
subject of non-Euclidean geometry and incidentally the 
idea of the fourth dimension. The great mass of 
non-professional mathematical readers, therefore, owe 
these men an immeasurable debt of gratitude for the 
work that they have done in the matter of rendering 
the conceptions which constitute the fabric of meta- 
geometry understandable and thinkable. A glance at 
the bibliography appended at the end of this volume 
will give some idea of the enormous amount of labor 
that has been expended in an effort to translate the 
most abstract mathematical principles into a language 
that could easily be comprehended by the average in- 
telligent person. 

The characteristic standpoint of -this period is the 
popular comprehension of the hyperspace concept and 
the consequent mental liberation which follows. For 
there is no doubt but that unheard of possibilities of 



SKETCH OF HYPERSPACE MOVEMENT 67 

thought have been revealed by investigations into the 
nature of space. An entirely new world has been 
opened to view and only a beginning has been made 
at the exploration of its extent and resources. 

One of the notable incidents of the early years of 
this period is the position taken by Felix Klein who 
stands in about the same relation to Cayley as 
Beltrami does to Riemann, in that he assumed the 
task of completing the work of his predecessor. 
Klein held that there are only two kinds of Rie- 
MANNIAN space — the elliptical and the spherical. Or 
in other words, that there are only two possible kinds 
of space in which the propositions announced by 
Riemann could apply. Sophus Lie, called the "great 
comparative anatomist of geometric theories," carried 
his classifications to a final conclusion in connection 
with spaces of all kinds and decided that there are 
possible only four kinds of three dimensional spaces. 

But whether men with peering, microscopic, histo- 
logical vision shall establish the existence of one or 
many spaces, and regardless of the mathematic rigor 
with which they shall demonstrate the self-consistency 
of the doctrines which they hold, the fact remains that 
the hypotheses thus maintained, while they may be 
regarded as true descriptions of the spaces concerned, 
are, nevertheless, incompatible. All of them cannot 
be valid. It will perhaps be found that none of them 
are valid, especially objectively so. The only true 
view, therefore, of these systems of hyperspaces is 
that which assigns them to their rightful place in the 
infinitely vast world of pure mathesis where their 
validity may go unchallenged and their existence un- 
questioned; for in that domain of unconfined menta- 



68 THE MYSTERY OF SPACE 

tion, in that realm of divine intuitability, the marvel- 
ous wonderland of ideas and notions, one is not only 
disinclined to doubt their logical actuality, but is quite 
willing to accede their claims. 



CHAPTER III 

Essentials of the Non-Euclidean Geometry 

The Non-Euclidean Geometry Concerned with Conceptual 
Space Entirely — Outcome of Failures at Solving the 
Parallel-Postulate — The Basis of the Non-Euclidean 
Geometry — Space Curvature and Manifoldness — Some 
Elements of the Non-Euclidean Geometry — Certainty, 
Necessity and Universality as Bulwarks of Geometry — 
Some Consequences of Efforts at Solving the Parallel- 
Postulate — The Final Issue of the Non-Euclidean Geom- 
etry — Extended Consciousness. 

The term "non-Euclidean" is used to designate 
any system of geometry which is not strictly Euclidean 
in content. 

It is interesting to note how the term came to be 
used. It appears to have been employed first by 
Gauss. He did not strike upon it suddenly, however, 
as in the correspondence between him and Wachter 
in 1816 he used the designation "anti-Euclidean" and 
then, later, following Schweikart, he adopted the 
latter' s terminology and called it "Astral Geometry." 
This he found in Schweikart's first published 
treatise known by that name and which made its ap- 
pearance at Marburg in December, 181 8. Finally, 
in his correspondence with Taurinus in 1824, Gauss 
first used the expression "non-Euclidean" to designate 
the system which he had elaborated and continued to 

69 



7 o THE MYSTERY OF SPACE 

use it in his correspondence with Schumacher in 
1831. 

"Non-Legendrean," "semi-Euclidean" and "non- 
Archimedean" are titles used by M. Dehn to denote 
all kinds of geometries which represented variations 
from the hypotheses laid down by Legendre, Euclid 
and Archimedes. - - 

The semi-Euclidean is a system of geometry in 
which the sum of the angles of a triangle is said to 
be equal to two right angles, but in which one may 
draw an infinity of parallels to a straight line through 
a given point. The non-Euclidean geometry embraces 
all the results obtained as a consequence of efforts 
made at finding a satisfactory proof of the parallel- 
postulate and is, therefore, based upon a conception 
of space which is at variance with that held by 
Euclid. According to the Ionian school space is an 
infinite continuum possessing uniformity throughout 
its entire extent. The non-Euclideans maintain that 
space is not an infinite extension; but a finite though 
unbounded manifold capable of being generated by 
the movement of a point, line or plane in a direction 
without itself. It is also held that space is curved 
and exists in the shape of a sphere or pseudosphere 
and is consequently elliptical. 

The inapplicability of Euclid's parallel-postulate 
to lines drawn upon the surface of a sphere suggested 
the possibility of a space in which the postulate could 
apply to all possible surfaces or that space itself may 
be spherical in which case the postulate would be in- 
validated altogether. Hence, it is quite natural that 
mathematicians finding themselves unable to prove the 
postulate with due mathetic precision should turn their 



THE NON-EUCLIDEAN GEOMETRY 71 

attention to the conceptually possible. In this virtual 
abandonment of the perceptual for the conceptual lies 
the fundamental difference between the Euclidean and 
the non-Euclidean geometries. It may be said to the 
credit of the Euclideans that they have sought to make 
their geometric conceptions conform as closely as 
possible to the actual nature of things in the sensuous 
world while at the same time they must have per- 
ceived that at best their spatial notions were only ap- 
proximations to the sensuous actuality of objects in 
space. 

On the other hand, non-Euclideans make no pre- 
tense at discovering any congruency between their 
notions and things as they actually are. The attitude 
of the metageometricians in this respect is very aptly 
described by Cassius Jackson Keyser who says: 

"He constructs in thought a summitless hier- 
archy of hyperspaces, an endless series of orderly 
worlds, worlds that are possible and logically actual, 
and he is content not to know if any of them be 
otherwise actual or actualized." 1 

The non-Euclidean is, therefore, not concerned 
about the applicability of ensembles, notions and 
propositions to real, perceptual space conditions. It 
is sufficient for him to know that his creations are 
thinkable. As soon as he can resolve the nebulosity 
of his consciousness into the conceptual "star-forms" 
of definite ideas and notions, he sits down to the feast 
which he finds provided by superfoetated hypotheses 
fabricated in the deeps of mind and logical actualities 

1 Mathematics, by C. J. Keyser, Adrian Professor of Mathematics, 
Columbia University. 



72 THE MYSTERY OF SPACE 

unperturbed and unmindful of the weal of perceptual 
space in its homogeneity of form and dimensionality. 

Fundamentally, the non-Euclidean geometry is con- 
structed upon the basis of conceptual space almost 
entirely. Knowledge of its content is accordingly 
derived from a superperceptual representation of re- 
lations and interrelations subsisting between and 
among notions, ideas, propositions and magnitudes 
arising out of a conceptual consideration thereof. In 
other words, representations of the non-Euclidean 
magnitudes, cannot be said to be strictly perceptual 
in the same sense that three-space magnitudes are 
perceived; for three-space magnitudes are really sense 
objects while hyperspace magnitudes are not sense ob- 
jects. They are far removed from the sensuous world 
and in order to conceive them one must raise his 
consciousness from the sensuous plane to the concep- 
tual plane and become aware of a class of perceptions 
which are not perceptions in the strict sense of the 
word, but superperceptions ; because they are repre- 
sentations of concepts rather than precepts. 

Notions of perceptual space are constituted of the 
triple presentations arising out of the visual, tactual 
and motor sensations which are fused together in their 
final delivery to the consciousness. The synthesis of 
these three sense-deliveries is accomplished by equi- 
librating their respective differences and by correct- 
ing the perceptions of one sense by those of another 
in such a way as to obtain a completely reliable per- 
ception of the object. This is the manner in which 
the characteristics of Euclidean space are established. 

The characteristics of non-Euclidean space are not 
arnved at exactly in this way. Being beyond the scope 



THE NON-EUCLIDEAN GEOMETRY 73 

of the visual, tactile and motor sense apprehensions, it 
cannot be said to represent judgments derived from 
any consideration or elaboration of the deliveries 
presented through these media. Yet, the substance of 
metageometry, or the science of the measurement of 
hyperspaces, may not be regarded as an a priori sub- 
structure upon which the system is founded. That is, 
the conceptual space of non-Euclidean geometry is not 
presented to the consciousness as an a priori notion. 
On the other hand, the a posterioristic quality of 
metageometric spaces marks the entire scope of motility 
of the notions appertaining thereto. 

The notions, therefore, of conceptual space are de- 
rivable only from the perception of concepts, or, 
otherwise consist of judgments concerning intercon- 
ceptual relations. The process of apperception in- 
volved in the recognition of relations which may be 
methodically determined is much removed from the 
primary procedure of perceiving sense-impressions and 
fusing them into final deliveries to the consciousness 
for conceptualization or the elaboration into concepts 
or general notions. It is a procedure which is in 
every way superconceptual and extra-sensuous. The 
metageometrician or analyst in no way relies upon 
sense-deliveries for the data of his constructions; for, 
if he did, he should, then, be reduced to the necessity 
of confining his conclusions to the sphere of motility 
imposed by the sensible world with the result that we 
should be able to verify empirically all his postula- 
tions. But, contrarily, he goes to the extra-sensuous, 
and there in the realm of pure conceptuality, he finds 
the requisite freedom for his theories; thus, environed 
by a sort of intellectual anarchism, he pursues ana- 



74 THE MYSTERY OF SPACE 

lytical pleasures quite unrestrainedly. The difference 
between the two mental processes — that which leads 
from the sensible world to conception and that which 
veers into the fields beyond — is so great that it is 
hardly permissible to view the results arrived at in 
the outcome of the separate processes as being 
identical. 

To illustrate this difference, let us draw an analogy. 
The miner digs the iron ore out of the ground. The 
iron is separated from the extraneous material and 
delivered to the furnaces where the metal is melted 
and turned out as pig iron. It is further treated, and 
steel, of various grades, cast iron and other kinds of 
iron are produced. The treatment of the iron ore 
up to this stage is similar to the treatment of sense- 
impressions by the Thinker. Steel, cast iron, et cetera, 
are similar to mental concepts. Later, the steel and 
other products are converted into instruments and 
numerous articles. This represents the superperceptual 
process. Trafficking in iron ore products, such as 
instruments of precision, watch springs, and the like, 
represents a stage still farther removed from the 
primary treatment of the ore and is similar to that 
to which concepts are treated when the metageometri- 
cian manipulates them in the construction of conceptual 
space-forms. Perception is the dealing with raw iron 
ore while conception is analogous to the production 
of the finished product. 

Superperception would be analogous to the traffick- 
ing in the finished product as such "and without any 
reference to the source or the preceding processes. 
Thus the notions and judgments of the non-Euclidean 
geometry are arrived at as a result of a triple process 



THE NON-EUCLIDEAN GEOMETRY 75 

of perception, conception and superperception the latter 
being merely superconceived as formal space-notions. 
But it is obvious that the more complex the processes 
by which judgments purporting to relate to perceptual 
things are derived the more likely are those judgments 
to be at variance with the nature of the things them- 
selves. 

In view of the foregoing, the dangers resulting 
from identifying the products of the two processes 
are very obvious indeed. But the difference between 
the two procedures is the difference between Euclidean 
and non-Euclidean geometries or the difference be- 
tween perceptual space notions and conceptual space 
notions. Hence, it is not understood just how or why 
it has occurred to anyone that the two notions could 
be made congruent. Magnitudes in perceptual, sen- 
sible space are things apart from those that may be 
said to exist in mathematical space or that space whose 
qualities and properties have no existence outside of 
the mind which has conceived them. It is believed to 
be quite impossible to approach the study of meta- 
geometrical propositions with a clear, open mind with- 
out previously understanding the fundamental distinc- 
tions which exist between them. 

It follows, therefore, as a logical conclusion that 
geometric space of whatsoever nature is a purely 
formal construction of the intellect, and for this rea- 
son is completely under the sovereignty of the intellect 
however whimsical its demands may be. Being thus 
the creature of the intellect, its possibilities are limited 
only by the limitations of the intellect itself. Per- 
ceptual space, being neither the creature of the intellect 
nor necessarily an a priori notion resident in the mental 



76 THE MYSTERY OF SPACE 

substructure, but existing entirely independent of the 
intellect or its apprehension thereof, cannot be ex- 
pected to conform to the purely formal restrictions 
imposed by the mind except in so far as those re- 
strictions may be determined by the nature of per- 
ceptual space. And for that matter, it should not be 
forgotten that, as yet, we have no means of determin- 
ing whether or not the testimony of the intellect is 
thoroughly credible simply because there is no other 
standard by which we may prove its testimony. It is 
possible to justify the deliveries of the eye by the sense 
of touch, or vice versa. It is also possible to prove 
all our sense-deliveries by one or the other of the 
senses. But we have no such good fortune with the 
deliveries of the intellect. We have simply to accept 
its testimony as final; because we cannot do any better. 
But if it were possible to correct the testimony of 
the intellect by some other faculty or power which 
by nature might be more accurate than the intellect 
it should be found that the intellect itself is sadly 
limited. 

The possible curvature of space is a notion which 
also characterizes the content of the non-Euclidean 
geometry. It is upon this notion that the question of 
the finity and unboundedness of space, in the mathe- 
matical sense, rests. In the curved space, the straight- 
est line is a curved line which returns upon itself. 
Progression eastward brings one to the west; pro- 
gression northward brings one to the south, et cetera. 
On this view space is finite, but may not be regarded 
as possessing boundaries. 

Space-curvature, reinforced by the idea that space 
is also a manifold is the enabling clause of meta- 



THE NON-EUCLIDEAN GEOMETRY 77 

geometry and without them the analyst dares not pro- 
ceed. Here again, we are led to the confession that 
however fantastic these two notions may seem and 
evidently are, there is nevertheless to be recognized in 
them a "dim glimpse" of a veritable reality — a slight 
foreshadowing of the revelation of some great kosmic 
mystery. 

The manifoldness of space is the fiat of analysis. 
It is the inevitable outcome of the analyst's method 
of procedure. His education, training and view of 
things in general inhibit his arriving at any other re- 
sult and he may be pardoned with good grace for his 
manufacture of the space-manifold. For by it per- 
haps a better appreciation of that wonderful extension 
of consciousness in the nature of which is involved the 
explanation of the perplexing problems which the 
manifold and other metageometrical expedients faintly 
adumbrate may be gained. 

It is pertinent, in the light of the above, to ex- 
amine into some of the relative merits of the three 
formal bulwarks of geometrical knowledge. These 
are certainty, necessity and universality. 

Geometric certainty is derived solely from the 
nature of the premises upon which it is based. If the 
premises be contradictory, it is, of course, defective. 
But if the premises are non-contradictory or self- 
evident, then the certainty of geometric notions and 
conclusions is valid. Another consideration of prime 
importance in this connection is the definition. From 
it all premises proceed. Hence, the definition is even 
more important than the premise; for it is the per- 
sisting determinant of all geometric conclusions while 
the premise is dependent upon the limitations of the 



78 THE MYSTERY OF SPACE 

definition. The determinative character of the defini- 
tion has led to its apotheosis; but this, admittedly, has 
been necessary in order to give stability and per- 
manency to the conclusions which followed. But in 
spite of this it would appear that the certainty of 
geometric conclusions is not a quality to be reckoned 
as absolute or final. 

With the same certainty that it can be said the 
sum of the angles of the triangle is equal to two right 
angles it may be asserted that that sum is also greater 
or less than two right angles. Certainty which is 
based upon the inherent congruity of definitions, 
premises and propositions is an entirely different 
matter from that certainty which arises out of the real, 
abiding validity of a scheme of thought. But this 
difference is not lessened by the fact that the latter 
is dependent, in a measure, upon the correct systema- 
tization of our spatial experiences by means of 
methodical processes. Euclidean geometry, accord- 
ingly, is not so certain in its applications as it is utili- 
tarian; but non-Euclidean geometry is even less certain 
than the former and consequently more lacking in its 
utilitarian possibilities. 

The necessity of geometrical determinations is 
merely the necessity which inheres in logical inferences 
or deductions. These may or may not be valid. 
Inasmuch as the necessariness of deductions is primarily 
based upon the conditional certainty of premises and 
definitions it appears that this quality is in no way 
peculiar to geometry whether Euclidean or non- 
Euclidean. In like manner, the universality of geo- 
metric judgments may not properly be regarded as 
a peculiarity of geometry; but is explicable upon the 



THE NON-EUCLIDEAN GEOMETRY 79 

basis of the formal character of the assumptions 
which underlie it. The chief value, then, of non- 
Euclidean geometry seems to abide in the fact that 
it clarifies our understanding as to the complex proc- 
esses by which it is possible to organize and sys- 
tematize our spatial experiences for assimilation and 
use in other branches of knowledge. 

With the above statement of the case of the 
non-Euclidean geometry it is now thought permissible 
to state briefly some of the elements thereof. 1 

Below will be found some of the elements ob- 
tained as a consequence of efforts made both at 
proving and disproving the parallel-postulate of 
Euclid : 

"If two points determine a line it is called a 
straight." 

"If two straights make with a transversal equal 
alternate angles they have a common perpendicular." 

"A piece of a straight is called a sect." 

"If two equal coplanar sects are erected per- 
pendicular to a straight, if they do not meet, then the 
sect joining their extremities makes equal angles with 
them and is bisected by a perpendicular erected mid- 
way between their feet." 

"The sum of the angles of a rectilineal triangle 
is a straight angle, in the hypothesis of the right 
(angle) ; is greater than a straight angle in the hypo- 



1 The science of pure mathematics is perhaps indebted to no one 
in so great a degree as to George Bruce Halstead, formerly of 
the University of Texas, whose labors in connection with the popu- 
lar exposition of the non-Euclidean geometry have been most un- 
tiring and effectual. Vide Popular Astronomy, Vol. VII and VIII, 
1900, Dr. G. B. Halstead. 



80 THE MYSTERY OF SPACE 

thesis of the obtuse (angle) ; is less than a straight 
angle in the hypothesis of the acute (angle)." 

"The hypothesis of right is Euclidean; the hypo- 
thesis of the acute is Bolyai-Lobachevskian; the 
hypothesis of obtuse is Riemannian." 

"If one straight is parallel to a second the second 
is parallel to the first." 

"Parallels continually approach each other." 

"The perpendiculars erected at the middle point 
of the sides of a triangle are all parallel, if two are 
parallel." 

"If the foot of a perpendicular slides on a straight 
its extremity describes a curve called an equidistant 
curve, or an equidistantial." 

"An equidistantial will slide on its trace." 

"In the hypothesis of the obtuse a straight is of 
finite size and returns into itself." 

"Two straights always intersect." 

"Two straights perpendicular to a third straight 
intersect at a point half a straight from the third either 
way." 

"A pole is half a straight from its polar." 

"A polar is the locus of coplanar points half a 
straight from its pole. Therefore, if the pole of one 
straight lies on another straight the pole of this second 
straight is on the first straight." 

"The cross of two straights is the pole of the join 
of their poles." 

"Any two straights inclose a plane figure, a digon." 

"Two digons are congruent if their angles are 
equal." 

"The equidistantial is a circle with center at the 
poles of its basal straight." 



THE NON-EUCLIDEAN GEOMETRY 81 

A typical postulate based upon the Bolyai hy- 
pothesis of the acute angle is the following: 

"From any point P drop PC, a perpendicular to 
any given straight line AB. If D move off indefinitely 
on the ray CB, the sect will approach as limit PF 
copunctal with AB at infinity. 




PD is said to be at P the parallel to AB toward 
B. PF makes with PC an angle CPF which is called 
the angle of parallelism for the perpendicular PC. 
It is less than a right angle by an amount which is 
the limit of the deficiency of the triangle PCD. On 
the other side of PC, an equal angle of parallelism 
gives the parallel P to BA towards AM. 3 Thus at 
any point there are two parallels to a straight. A 
straight has, therefore, two separate points at infinity." 

"Straights through P which make with PC an angle 
greater than the angle of parallelism and less than its 
supplement do not meet the straight AB at all not 
even at infinity." 

8 Note.— M may be any point on the line BA indefinitely produced. 



82 THE MYSTERY OF SPACE 

The parallel-postulate is stated in the non- 
Euclidean geometry as follows: 

"If a straight line meeting two straight lines make 
those angles which are inward and upon the same side 
of it less than two right angles the two straight lines 
being produced indefinitely will meet each other on 
this side where the angles are less than two right 
angles." 

It is stated by Manning 4 in the following lan- 
guage: 

"If two lines are cut by a third and the sum of 
the interior angles on the same side of the cutting 
line is less than two right angles the line will meet 
on that side when sufficiently produced." 

It is rather significant that in this postulate which 
is really a definition of space should be found grounds 
for such diverse interpretations as to its nature. Of 
course, the moment the mind seeks to understand the 
infinite by interpreting it in the unmodified terms of the 
apparently unchangeable finite it entangles itself into 
insurmountable difficulties. As a drowning man grasps 
after straws so the mind, immersed in endless abysses 
of infinity, fails to conduct itself in a seemly manner; 
but gasps, struggles and flounders and is happy if it 
can, in the depths of its perplexity, discover a way of 
logical escape. The pure mathematician has a hanker- 
ing after the logically consistent in all his pursuits; to 
him it is the "Holy Grail" of his highest aspirations. 
He seeks it as the devotee seeks immortality. It is to 
him a philosopher's stone, the elixir of perpetual youth, 
the eternal criterion of all knowledge. 

Failures to demonstrate the celebrated postulate of 
•Vide Non-Euclidean Geometry, p. 91. 



THE NON-EUCLIDEAN GEOMETRY 83 

Euclid led, as a matter of course, to the substitution 
of various other postulates more or less equivalent to 
it in that each of them may be deduced from the 
other without the aid of any new hypothesis. 

Among those who sought proof by a restatement 
of the problem are the following: 

1. Ptolemy: The internal angles which two par- 
allels make with a transversal on the same side are 
supplementary. 

2. Clavius : Two parallel straight lines are equi- 
distant. 

3. Proclus: If a straight line intersects one of 
two parallels it also intersects the other. 

4. Wallis: A triangle being given another tri- 
angle can be constructed similar to the given one and 
of any size whatever. 

5. Bolyai (W.) : Through three points not lying 
on a straight line a sphere can always be drawn. 

6. Lorenz: Through a point between the lines 
bounding an angle a straight line can always be drawn 
which will intersect these two lines. 

7. Saccheri: The sum of the angles of a tri- 
angle is equal to two right angles. 

There were, of course, many other statements and 
substitutions used by mathematicians in their en- 
deavors satisfactorily to establish the truth of the 
parallel-postulate. That their labors should have 
terminated, first, by doubting it, then by denying, and 
finally, by building up a system of geometries which 
altogether ignores the postulate is just what might 
naturally be expected of these men who have given 
to the world the non-Euclidean geometry. In doing 
what they did many, if not all of them, were not aware 



84 THE MYSTERY OF SPACE 

in any measure of the proportions of the imposing 
superstructure that would be built upon their apparent 
failures. All of them undoubtedly must have sensed 
the vague adumbrations forecast by the unfolding mys- 
teries which they sought to lay bare; all of them must 
have felt as they executed the early tasks of those 
crepuscular days of pure mathematics that the way 
which they were traveling would lead to the inner 
shrine of a higher knowledge and a wider freedom; 
they may have been led by divine intuition to strike 
out on this new path and yet they could not have 
known how fully their dreams would be realized by the 
mathematicians of the twentieth century. If so, they 
were truly gods and mathesis is their kingdom. 

The analyst proceeds upon a basis entirely at vari- 
ance with that which guides the ordinary investigator 
in the formulation of his conclusions. The empirical 
scientist in arriving at his theories or hypotheses is 
governed at all times by the degree of conformity 
which his postulates exhibit to the actual phenomena 
of nature. He endeavors to ascertain just how far 
or in what degree his hypothesis is congruent with 
things found in nature. If the dissidence is found to 
predominate he abandons his theory and makes another 
statement and again sets out to determine the degree 
of conformity. If he then finds that the natural phe- 
nomena agree with his theory he accepts it as for the 
time being finally settling the question. In all things 
he is limited by the answer which nature gives to his 
queries. Not so with the exponent of pure mathe- 
matics. For him the truth of hypotheses and postu- 
lates is not dependent upon the fact that physical 
nature contains phenomena which answer to them. The 



THE NON-EUCLIDEAN GEOMETRY 85 

sole determining factor for him is whether or not Re 
is able to state with rational consistency the assumed 
first principles and then logically develop their conse- 
quences. If he can do this, that is, if he can state his 
hypotheses with consistency and develop their conse- 
quences into a logical system of thought, he is quite 
satisfied and well pleased with his performances. But 
the fact that this is true is of vital significance for all 
who seek clearly to understand the essential character 
of hyperspatiality. 

It appears, therefore, that the science of conse- 
quences is the radical essence of pure geometry. The 
metageometrician enjoys unlimited freedom in the 
choice of his postulates and suffers curtailment only 
when it comes to the question of consistency. He is 
at liberty to formulate as many systems of geometry as 
the barriers of consistency will permit and these are 
practically innumerable. So long then as the laws of 
compatibility remain inviolate his multiplication of 
postulate-systems may proceed indefinitely. Is it 
strange then that under conditions where an investi- 
gator has such unbridled liberty he should be found 
indulging in mathetic excesses? 

Kant held that the axioms of geometry are syn- 
thetic judgments a priori; but it appears that in the 
strictest sense this is not the case. It depends upon 
the type of mind which is taken as a standard of refer- 
ence. If it be the uncultivated mind, it is certain that 
to it the relations expressed by an axiom would never 
appear spontaneously. If on the other hand, the 
standard be that of a cultivated mind it is also equally 
certain that to it these relations would be discovered 
only after methodical operations. All judgments ar- 



86 THE MYSTERY OF SPACE 

rived at as a result of logical processes should, it seems, 
be regarded as judgments a -posteriori, i.e., the results 
of empirical operations. Confessedly, the facts ad- 
duced in course of experimentation serve as guides in 
choosing among all of the many possible logical conven- 
tions; but our choice remains untrammeled except by 
the compulsion arising out of a fear of inconsistency. 
The real criterion then of all geometries is neither 
truth, conformability nor necessity, but consistency and 
convenience. 

The difficulty with the non-Euclideans resolved 
itself into the question as to whether it is more con- 
sistent, as well as convenient, to establish a proof of 
the postulate by taking advantage of the support to 
be found in other postulates or whether, by seeking a 
demonstration based upon the deliveries of sense-ex- 
perience as to the nature of space and its properties, 
a still more consistent conclusion might be reached. 
They had further perplexity, however, when it came 
to a decision as to whether the organic world is pro- 
duced and maintained in Euclidean space or in a purely 
conceptual space which alone can be apprehended by 
the mind's powers of representation. Unwilling to 
admit the existence of the world in Euclidean space, 
they turned their attention to the examination of the 
properties of another kind of space so-called which 
unlike the space of the Ionian school could be made to 
answer not only all the purposes of plane and solid 
figures, but of spherics as well. And so, the manifold 
space was invented by Riemann and later underwent 
some remarkable improvements at the hands of his 
disciple, Beltrami. But it may be said here, paren- 
thetically, that the truth of the whole matter is that 



THE NON-EUCLIDEAN GEOMETRY 87 

our world is neither in Euclidean nor non-Euclidean 
space, both of which, in the last analysis, are concep- 
tual abstractions. Although it may not be denied that 
the Euclidean space is the more compatible. 

The problem of devising a space, if only a very 
limited portion, in which could be demonstrated the 
assumed alternative hypothesis and its consequences 
logically developed, occasioned no inconsiderable con- 
cern for the non-Euclidean investigators ; but neither 
Lobachevski, Bolyai nor Riemann were to be 
baffled by the difficulties which they met. These only 
cited them to more laborious toil. Having succeeded 
in mentally constructing the particular kind of space 
which was adaptable to their rigorous mathetic re- 
quirements it immediately occurred to them that all the 
qualities of the limited space thus devised might 
logically be amplified and extended to the entire world 
of space and that what is true of figures constructed 
in the segmented portion of space which they used 
for experimental purposes is also true of figures drawn 
anywhere in the universe of this space as all lines 
drawn in the finite, bounded portion could be extended 
indefinitely and all magnitudes similarly treated. From 
these results, it was but a single step to the conclusion 
which followed — that either an entirely new world 
of space had been discovered or that our notion of 
the space in which the organic world was produced is 
wholly wrong and needs revision. But notwithstand- 
ing the insurmountable obstacles which stood in the 
way of the investigators who made the attempt to 
discover the homology which might exist between the 
characteristics of the newly fabricated space and the 
phenomenal world, investigations were carried for- 



88 THE MYSTERY OF SPACE 

ward with almost amazing recklessness and loyalty to 
the mathetic spirit until it was discovered that all efforts 
to trace out any definite lines of correspondence were 
futile. Then the policy of ignoring the question of 
conformability was adopted and has since been pur- 
sued with unchangeable regularity by the analytical 
investigator. 

Among the results obtained by the non-Euclideans 
in their profound researches into the nature of hyper- 
space are these: i. It was found that the angular sum 
of a triangle, being ordinarily assumed to be a variable 
quantity, is either less or greater than two right angles 
so that a strictly Euclidean rectangle could not be con- 
structed. 2. The angle sums of two triangles of equal 
area are equal. 3. No two triangles not equal can 
have the same angles so that similar triangles are im- 
possible unless they are of the same size. 4. If two 
equal perpendiculars are erected to the same line, their 
distance apart increases with their length. 5. A line 
every point of which is equally distant from a given 
straight line is a curved line. 6. Any two lines which 
do not meet, even at infinity, have one common per- 
pendicular which measures their minimum distance. 
7. Lines which meet at infinity are parallel. But it is 
apparent that these results have not followed upon any 
mathematical consequence of other supporting postu- 
lates or axioms such as would place them on a coordi- 
nate basis with those used as a support for the parallel- 
postulate; for they are based upon the envisagement of 
an entirely new principle of space-perception and be- 
long to a wholly different set of space qualities. 

The final issue then of the non-Euclidean geometry 
is neither in the utility of its processes and conclusions 



THE NON-EUCLIDEAN GEOMETRY 89 

nor in the increscent inclination towards a new outlook 
upon the world of mathesis; but resides solely in the 
possibilities yet to be developed in that vast domain of 
analytical thought which it has discovered and opened 
to view. To say that it sheds any light upon the nature 
of the universe is perhaps to take the radical view; 
yet it cannot be doubted that the researches incident 
to the formulation of the non-Euclidean geometry 
have greatly extended the scope of consciousness. 
Whether the extension is valid and normal or simply 
a hypertrophic excrescence of mental feverishness; 
whether by virtue of it we shall more closely approach 
an understanding of the true nature of the mind of 
the Infinite, or shall all fall into insanity, are certainly 
debatable questions. It nevertheless appears evident 
that humanity has gained something of real, abiding 
permanence by this new departure. If that something 
be merely an extended consciousness or an awakening 
to the fact that there are stages of awareness beyond 
the strictly sensuous, and every observable evidence 
points to this, then there has only begun the process 
by which the faculty of conscious functioning in this 
new world shall become the normal possession of the 
human species. But this new world cannot be said to 
be of mathematical import; for it is doubtful if mathe- 
matical laws such as have been devised up to the 
present time, would obtain therein. So that if any- 
thing, it must be psychological and vital. 

On this view the worlds of hyperspace inlaid with 
analytic manifoldnesses and constant curvatures are 
but the primal excitants which will finally awaken in 
the mind the faculty of awareness in the new domain 
of psychological content. Then will come the bloom- 



90 THE MYSTERY OF SPACE 

ing of the diurnal flower of the mind's immortality and 
the outputting of the organ of consciousness where- 
with the infinite stretches of hyperspaces, the low-lying 
valleys of reals and imaginaries and the uplifting 
hills of finites and infinites shall be divested of their 
mysteries and stand out in their unitariness no longer 
draped in the veil of the inscrutable and the incompre- 
hensible. 

The fourth dimension, regarded by some as a new 
scope of motion for objects in space, by others as a 
new and strange direction of spatial extent and by 
others still as the doorway of the temple of exegesis 
wherein an explanation may be found for the entire 
congeries of mysteries and supermysteries which now 
perplex the human mind, may also be said to be 
the key to the non-Euclidean geometry. But it really 
complicates the situation; for one has to be capable of 
prolonged abstract thought even to envisage is as a 
conceptual possibility. Poincare 5 says: "Any one 
who should dedicate his life to it could, perhaps, 
eventually imagine the fourth dimension,'' implying 
thereby that a lifetime of prolonged abstract thought 
is necessary to bring the mind to that point of ecstasy 
where it could even so much as imagine this additional 
dimension. Nevertheless by it (the fourth dimension) 
was the non-Euclidean geometry made and without it 
was not any of the hyperspaces made that were made. 
It is the view which geometers have taken of space 
in general that has made the fourth dimension pos- 
sible, and not only the fourth, but dimensions of all 
degrees. The basis of the non-Euclidean geometry 

•Vide Nature, Vol. XLV, 1892. 



THE NON-EUCLIDEAN GEOMETRY 91 

may be found then in the notion of space which has 
been predominant in the minds of the investigators. 

Finally, it should be pointed out that the non- 
Euclidean geometry, though a consistent system of 
postulates, has been constructed upon a misconception 
based upon the identification of real, perceptual space 
with systems of space-measurements. Hyperspaces 
which are not spaces at all should not be confounded 
with real space. But they constitute the substance of 
non-Euclidean geometry; they are its blood and sinews. 
Their study is interesting, because of the possibilities 
of speculation which it offers. No mind that has 
thought deeply upon the intricacies of the fourth di- 
mension, or hyperspace, remains the same after the 
process. It is bound to experience a certain sense of 
humility, and yet some pride born of a knowledge that 
it has been in the presence of a great mystery and has 
delved into the fearful deeps of kosmic mind. To the 
mind that has thus been anointed by the sacred chrism 
of the inner mysteries of creative mentality there 
always come that stillness and calm such as character- 
ize the aftermath of reflection upon the incompre- 
hensible and the transfinite. 



CHAPTER IV 

Dimensionality 

Arbitrary Character of Dimensionality — Various Definitions of 
Dimension — Real Space and Geometric Space Differen- 
tiated — The Finity of Space — Difference Between the 
Purely Formal and the Actual — Space as Dynamic Appear- 
ance — The A Priori and the A Posteriori as Defined by 
Paul Carus. 

In previous chapters we have traced the growth 
and development of the non-Euclidean geometry show- 
ing that the so-called fourth dimension is an aspect 
thereof. It is now deemed fitting that we should 
enter into a more detailed study of the question of 
dimensionality with a view to examining some of the 
difficulties which encompass it. 

The question of dimension is as old as geometry 
itself. Without it geometric conclusions are void and 
meaningless. Yet the conception of dimensionality 
itself is purely conventional. In its application to space 
there is involved a great deal of confusion because of 
the inferential character of its definition. For instance, 
commonly we measure a body in space and arbitrarily 
assign three elements to determine its position. The 
simplest standard for this purpose is the cube having 
three of its edges terminating at one of its corners. 

Thus because it is found that the entire volume of 
a cube is actually comprehended within the directions 

92 



DIMENSIONALITY 



93 



indicated by the lines ab t be and db it is determined 
that the three coordinates of the point b are necessary 
and sufficient to establish the dimensions of the cube 
and consequently of the space in which it rests. The 
conception may be stated in this way: If a collection 
of elements, say points or lines, be of such a nature or 
order that it is sufficient to know a certain definite num- 
ber of facts about it in order to be able to distinguish 
every one of the elements from all the others, then 




Fig. 6. 



the assemblage or collection of elements is said to be 
of the same number of dimensions as there are ele- 
ments necessary to its determination. In the above 
figure there are three elements, namely, the lines ab, 
be, and db, which are necessary and sufficient for the 
determination of the position of the point b. In this 
way geometers have determined that our space is tri- 
dimensional; but it is obvious that this conclusion is. 
based not upon any examination of space itself but 
upon the measurement of bodies in space. Upon this 
view it is seen that conclusions based upon such a pro- 
cedure render our notion of the extension of bodies in 



94 THE MYSTERY OF SPACE 

space identical with the notion of spatial extensity. In 
other words, we take bodies in space and by examining 
their characteristics and properties arrive at an al- 
leged apodeictic judgment of space. It is by means 
of this conventional norm of geometric knowledge that 
various other spaces, notably the one-, two-, four- and 
«-space, have been devised. It would appear that if 
some more absolute standard of measurement or defini- 
tion of space were adopted the confusion which now 
clings to the conception of dimension could be obviated. 
For if it be true that three and only three elements are 
necessary to determine a point-position in our space 
and that in this determination we also find the number 
of dimensions of space, then it may also be true that 
w-coordinates would just as truly determine the di- 
mensionality of an w-space, which is granted. But then 
the w-space would be just as legitimate as the three- 
space; for it is determined by exactly the same stand- 
ards. It is both quantitatively and qualitatively the 
same. If, however, on account of the exigencies that 
might arise, we are forced to seek solace in the notion 
of an w-space whither shall we turn for it? It cannot 
be found; for it is imperceptible, uninhabitable, non- 
existent, and therefore, absolutely and purely an ab- 
straction. Consequently, there must be something 
radically wrong with the definition of space or with 
its determinants. 

The purely arbitrary character of dimensionality is 
very aptly described by Cassius JACKSON Keyser, 
who says: 

". . . The dimensionality of a given space is 
not unique, but depends upon the choice of the 
geometric entity for primary or generating element. 



DIMENSIONALITY 95 

A space being given, its dimensionality is not there- 
with determined, but depends upon the will of the 
investigator who by a proper choice of generating 
element endows the space with any dimensionality 
he pleases. That fact is of cardinal significance for 
science and philosophy." x 

It is a fact of "cardinal significance" for science; 
because it emphasizes the necessity for some more 
rational procedure than that of the geometrician in ar- 
riving at an absolutely unique method of determining 
the dimension and essential nature of real space. Its 
significance for philosophy lies in the need of a logical, 
rigidly exclusive and absolutely peculiar standard of 
space definition. The definition of perceptual space 
should be such as rigorously inhibits its inclusion as a 
particular in any general class. The necessity for this 
is warranted by its universality and uniqueness. 

The lines of demarkation between what is recog- 
nized as perceptual space and what has been called 
geometric or conceptual space should be very sharply 
drawn. So that when reference is made to either there 
will be no doubt as to which is meant. And then, too, 
conceptual space is no space at all, properly speaking. 
It is merely a system of space-measurement. And as 
such has no logical right to be put in the same category 
as perceptual space. 

Real space is unique. Geometric space belongs to 
a class whose members are capable of indefinite multi- 
plication. It is certainly most illogical to identify 
them. Perceptual space, figuratively speaking, is a 
quantity; analytic space is the foot-rule, the yard-stick, 
the kilometer, by which it is measured and apportioned. 

1 Vide Monist, Vol. XVI, 1896, Mathematical Emancipations. 



96 THE MYSTERY OF SPACE 

It is logically impossible to predicate the same con- 
clusion for both of them. That is, to do so causes a 
profound fracture of the fundamental norms of logic. 
Such conclusions being thus illegitimate it is rather sur- 
prising that an error of this nature should have been 
made. It is perhaps accountable for on the grounds of 
the geometer's complete insouciance as to how his 
postulates shall stand in their relation to things in the 
phenomenal world. 

It is agreed that as convenient as is Euclid's system 
of space-measurement it is not by any means con- 
gruent with the extension of real space objects. It 
does, however, approximate congruity with these ob- 
jects as nearly as possible. How then could it be ex- 
pected that a system of space-measurement so far 
removed from this primary congruence as the non- 
Euclidean system is should exhibit more obvious signs 
of correspondence? But the advocates of the w-dimen- 
sionality of space have illatively asserted the identity 
of space and its dimensions. Accordingly, there is not 
recognized any distinction between their conception of 
space itself and its qualitative peculiarities. They use 
the terms interchangeably. So that dimension means 
space and vice versa. In this lack of discrimination 
may be found the source of much of the confusion 
which attaches to the conception of space. 

If it were arguable that the relation between space 
and its dimensions is the same as that between matter 
and its properties then the restriction of this relation to 
three and only three directions of extent would be dis- 
allowed; for the reason that if, as is commonly done, 
dimension be made to mean direction of extent, there 
would be an unlimited number of directions of extent 



DIMENSIONALITY 97 

and they would all be perceptible. But this is really 
another fundamental fault. Non-Euclideans have 
stretched the meaning of the term dimension so that 
it not only includes the idea of direction but an entirely 
new class of qualities — the fourth dimension. And 
despite this reformation of the original conception, 
they demand that it shall be called space. 

We have just shown that the generic concept of 
dimensionality is that three and only three coordinates 
are necessary and sufficient for its determination. 
Granting that this is true, are we not compelled con- 
sequently to see that we have, by adding a fourth or 
^-dimensions, involved ourselves into a more complex 
situation than before? For by postulating a fourth 
dimension either we have created a new world whose 
dimensions are four in number or we have explicitly 
admitted that the three dimensions have a fourth. 
Aside from the logical difficulties which beset these con- 
clusions there is also set up a condition which is at 
variance with the most elementary requirements of 
common sense. 

Thus far mathematical thought has not served to 
clarify our notions of space nor to shed any new light 
upon the vital processes which are alleged to have 
their explanation in the new discovery. Simply stated, 
metageometricians have brought us to the place where 
we must either recognize that the fourth dimension is 
another sphere lying dangerously near the earth in 
which space extends in four primary directions and in 
which four coordinates are necessary for its determina- 
tion or we are driven to the other horn of the dilemma 
where we are brought face to face with the conclusion 
that the three perceptual space dimensions have in 



98 THE MYSTERY OF SPACE 

common a hitherto unknown property or extension in 
virtue of which it may be viewed as having an un- 
limited number of dimensions. To accept the latter 
view is equivalent to saying that, in the above figure, 
the three lines ab, be and db have formed a triple 
entente by which they have mutually and severally ac- 
quired a new domain, hyperspace, and in which, be- 
cause of the vast resources of the region, they are able 
to perform wondrous things. 

Let us examine briefly the various current defini- 
tions of dimension. It is assumed by not a few that 
dimension is the same as direction. But can we grant 
this wholly to be true? If so, then a mere child may 
see that there are and must necessarily be as many 
dimensions as there are directions. Primarily, there are 
six directions of space and an unlimited number of 
subsidiary directions. On this view it is not necessary 
to invent a new domain of space if the object be 
merely to discover and utilize a greater number of 
dimensions than has heretofore been allowed. For 
the identification of the term dimension with direction 
already makes available an almost infinite number of 
dimensions. But this view is objected to by the advo- 
cates, for it is contrary to the hypothesis of «-dimen- 
sionality. 

Dimension also means extent. This is partially 
true. It cannot be wholly true. For, if it were, then, 
space would have only one dimension which is also not 
allowable under the hypothesis. Then the definition 
leaves out of account the idea that space is at the same 
time a direction or collection of directions. The term 
extension is generic and when applied to space means 
extension in all possible directions and not in any one 



DIMENSIONALITY 99 

direction. So that it is not permissible to say that 
space extends in this direction or that because it ex- 
tends in all directions simultaneously and equally. 

Geometers claim that space is a system of coordi- 
nates necessary for the establishment of a point-posi- 
tion in it. This view, however, identifies space with 
a system of space-measurement and is therefore faulty. 
According to this view there may be as many spaces 
as there are systems of space-measurement and the 
latter may be limitless. But if the totality of spaces 
are to be viewed as one space then we shall have one 
space with an indefinite number of dimensions ; also an 
indefinite number of space measurements which would 
be confusing. Much, if not all, of such a system's 
utility and convenience would be unavailable or use- 
less. That, too, would be in violation of the avowed 
purpose of these investigations which is to enhance the 
utility and convenience of mathematic operations. 

Now it is evident that space is neither direction, 
extension, a system of space-measurement nor a system 
of manifolds whose dimensions are generable. And 
this is so for the same reason that a piece of cloth 
is not the elements of measurement — inches, feet, 
yards — by which it is apportioned. And because we 
find that the fabric of space lends itself accommoda- 
tingly to our conventional norms of measurement is 
not sufficient reason for identifying it with these norms. 
Here we have the source of all error in mathematical 
conclusions about the nature of space; because all such 
conclusions are based not upon the intrinsic nature^# 
space, but upon artificial forms which we choose to 
impose upon it for our own convenience. But it should 
be remembered that the irregularities which we note 



ioo THE MYSTERY OF SPACE 

are not in space itself but inhere in the forms which we 
use. For these purposes space is extremely elastic and 
accommodates itself to the shape and scope of any 
construction we may decide to try upon it. In this 
respect it is like water which has no regard for the 
shape, size or kind of vessel into which it may be 
posited. There is one thing certain that judging from 
the above considerations there has been not yet any 
absolute, all-satisfying definition devised for space by 
mathematicians. 

The best definitions hitherto constructed are purely 
artificial and arbitrary determinations. It is rather 
anomalous that there should be so little unanimity 
about what is the most fundamental consideration of 
mathematical conclusions which are supposed to be so 
certain, so necessary and universal as to be incontro- 
vertible. Confessedly, it is a condition which raises 
again the question as to just what are the limits of 
mathematical certainty and necessity and just how far 
we shall depend upon the validity of mathematics to 
determine for us absolutely certain conclusions about 
the nature of space. In view of the uncertainty noted, 
are we justified in following too closely the mathematic 
lead even in matters of logic, to say nothing of our 
conception of space? It seems that we shall have 
necessarily, on account of the recognized limitations of 
mathematics in this matter, to turn to some more 
tenable source for the norms of our knowledge con- 
cerning space. For in the light of the rather inde- 
finable position which metageometricians have in- 
volved themselves there appears to be no hope in this 
direction. 

It is undoubtedly safer not to rely altogether upon 



DIMENSIONALITY 101 

the purely abstract, even in the world of mathesis, 
for any absolute criterion of knowledge. It is per- 
haps well that we should expunge the word absolute 
from our vocabularies. It is really a misnomer and 
has no meaning in the lexicon of nature. There is in 
reality no absolute in the sense of final absolution from 
all conditions or restrictions. 

In the ultimate analysis there is unquestionably no 
hue, tone, quality, condition nor any imaginable pos- 
ture of life, being or manifestation that is absolved 
from every other one of its class or from the totality. 
All these are relational and interdependent. There 
is no room for the absolute. In fact, it is a quality 
which cannot in any way be ascribed to any aspect of 
kosmic manifestation. It has existence only in the 
mind and has been devised for the purpose of mark- 
ing the limits of its scope. All being is relative; all 
life is relative and is destined to change its qualities 
as it evolves. All knowledge is also relative and what 
is true of one state may not be true of another; what 
is true of one life may not be true of another life; 
the limitations of one degree of knowledge may not 
have any bearings upon another degree. The norms 
of one will not satisfy the conditions of another stage 
of manifestation. It is always within limits that the 
criterion of knowledge will be found to satisfy a given 
set of conditions. Hence within certain limits mathe- 
matical conclusions will maintain their validity. Error 
is committed by pushing the validity of these limits to 
a position without the sphere of limitations. This 
seems to be the crux of the whole matter. Mathe- 
maticians, notably non-Euclideans, have sought to ex- 
tend the comparatively small sphere of limits of con- 



102 THE MYSTERY OF SPACE 

gruence between mathematic and perceptual space to 
such an extent as to cause it to encroach upon forbidden 
territory. In doing this they have erred grievously, 
causing serious offense to the more sensitive spirit of 
the high-caste mathematicians among whom are 
none more truly conservative than Paul Carus, 2 
who says: 

"Metageometricians are a hot-headed race and 
display sometimes all the characteristics of sectarian 
fanatics. To them it is quite clear there may be two 
straight lines through one and the same point which 
do not coincide and yet are both parallel to a third 
ine." 

To the student who has carefully followed the de- 
velopment of the non-Euclidean geometry and the 
notion of hyperspace the above characterization is 
none too severe nor ill-deserved. Nothing could more 
vividly yet correctly portray the impious tactics of the 
metageometrician and establish his perceptual obliquity 
more surely than the mere fact, mentioned by 
Carus, that he can with evident lack of mental per- 
turbation proclaim that two straight lines, noncoinci- 
dent with each other, may pass through a point and 
yet be parallel to a third line. But this is a mere trifle, 
a bagatelle, to the many other infractions of which 
he is guilty. The wonder is that he is able to secure 
such obsequious acceptance of his offerings as many of 
the most serious minded mathematicians are inclined 
to give. Is it to be wondered at that, despite the 
profuse protestations of the advocates, many who 
take up the study of the question of hyperspace should 
'Vide Monist, Vol. XIX, p. 402 (1009). 



DIMENSIONALITY 103 

experience a deep revulsion from the posture assumed 
by metageometricians with respect to these queries? 

Linked with the idea of dimensionality is the 
notion that space is infinite. This is a conception which 
has its roots imbedded in the depths of antiquity. 
Primitive man, looking up into the heavens at what 
appeared to him as a never ending extension, was 
awed by its vastness; but the minds of the most 
learned of the present-day men are not free from this 
innate dread of infinity. It permeates the thought life 
of all alike and none seems to be able to rise above it. 
Mathematicians, philosophers, scientists all share in 
the general belief that space is without limit, unend- 
ing in extent and eternally existent. Riemann, whose 
thought life found its most convenient mode of expres- 
sion by means of pure mathematics, was the first in the 
history of human thought to surmise that space is not 
infinite but limited even though unbounded. But his 
conception has been much vitiated on account of its 
entanglement with an idealized construction by which 
space is regarded as a thing to be manipulated and 
generated by act of thought. Were it not for this his 
conception would indeed mark the beginning of a new 
era in psychogenesis. As it is, when all the nonsensical 
effusions have been cleared away from our space con- 
ceptions and men come really to understand something 
of the essential nature of space this new era will find 
its true beginnings in the mind of Riemann. Although 
it must be said, as is the case with all progressive 
movements, the later development of a rationale for 
this conclusion will vary greatly from his original con- 
ception. For he had in mind a space that is generable 
and therefore a logical construction while ultimately 



io 4 THE MYSTERY OF SPACE 

the mind will swing back to a consideration of real 
space. 

Already men are beginning to see a new light. 
Already they are beginning to take a new view of space 
in general. The departure is especially noticeable 
in the attitude assumed by Hiram M. Stanley. 8 
He says: 

"If we seek the most satisfactory understand- 
ing of space we shall look neither to mathematics 
nor Psychology but to Physics. The trend of 
Physics, say with such a representative as Ostwald, 
is to make things the expression of force; the con- 
stitution and appearance of things are determined 
by dynamism; and we may best interpret space as a 
mode of this dynamic appearance." 

Space, as a mode of dynamic appearance is a slight 
improvement upon the old idea of a pure vacuity; for 
in the light of what we now know about space con- 
tent much of the dignity of that view has been lost. 
Men now know that space is not an empty void. They 
know that the atmosphere fills a great deal of space. 
They also have extended their conception in this direc- 
tion to include the ether and occultism goes further and 
postulates four kinds of ether — the chemical, life, light 
and psychographic ethers. But it does not stop here. 
It postulates a series of grades of finer matter than 
the physical which fills space and permeates its entire 
extent even to identification with its essential nature. 

Stanley continues: 

"Everything does not, as commonly conceived, 
fall into some pre-existent space convenient for it; 

• Philosophical Review, Vol. VII (1898). 



DIMENSIONALITY 105 

but everything makes its own spaciousness by its own 
defensive and offensive force, and the totality of all 
appearance is space in general." 

According to Stanley, not only do physical, per- 
ceptual objects, by their "offensive and defensive 
force" make their own space but the appearance of 
that in which no physical object is makes room for 
itself by its own dynamic force. In other words, that 
which we call "pure extensity" is by virtue of its 
dynamism the cause of its own existence. 

At first hand there appears to be little worthy of 
serious consideration in this view of Stanley; yet, 
if carried to its logical conclusion, the merit of the 
hypothesis becomes apparent. Accordingly, inter- 
stellar distances which are commonly said to be even 
without air or life of any kind are really an appearance 
possessed of a dynamism peculiar to itself. And this 
very force-appearance, constituting space, is that which 
makes it perceivable. For instance, let us say the space 
that exists between the earth and the moon, is not 
really empty nor does it have an existence prior to itself, 
but is a mode of dynamic appearance which is the cause 
of its own existence. Its dynamic character makes it to 
appear perceptible to our senses. Logically, if the 
dynamism were removed there would remain neither 
space nor the appearance of space. If this were true, 
and it is worthy of serious thought, then space is cer- 
tainly finite, as in its totality, according to Stanley's 
view, it would have to be regarded as a "phenomenon 
of the inner and finite life of the infinite." / 

It is believed that we may go a step further and 
unqualifiedly assert that space is finite, even denying 
its infinity as a "general mode of the activity of the 



io6 THE MYSTERY OF SPACE 

whole." Yet it is transfinite in the sense that it tran- 
scends the comprehension of finite minds or processes. 
It is finite because it is in manifestation. Everything 
that is in manifestation is finite. The infinite is not in 
manifestation. Infinity has to be limited always to 
become manifest. The Deity has limited His being in 
order that there may be a manifested universe. All 
things, all appearances are finite; because they are 
phenomena connected with manifestation. 

This question may be viewed from another stand- 
point. All things in manifestation or existence are 
polar in their constitution. For instance : there cannot 
be a "here" without a "there." There cannot be an 
"upper" without a "lower." Right is copolar with 
wrong; good is copolar with evil; night with day; 
manifestation with non-manifestation; truth with 
falsity; infinity with finity and so on, throughout the 
whole gamut of the pairs of opposites. What is the 
logical inference? Space is paired with a lack of 
space. There cannot be what we call space without 
there being at the same time the possibility, at least, 
of the lack of space or spacelessness. This is a con- 
clusion that is rigorously logical and incontrovertible. 

But it has been urged that it is impossible for the 
mind to imagine a condition where there is no space. 
It even has been asserted that it is contrary to the 
constitution of the mind itself to imagine "no space." 
But whether imaginable or not has no effect whatever 
upon the validity of the conception. Neither, it is said, 
can we imagine a fourth dimension but the mind has 
come dangerously near to imagining it. The distance 
from excogitating upon, discussing and describing the 
properties of four-space to imagining it is not so great 



DIMENSIONALITY 107 

after all. Truly it is difficult indeed, it seems, to be 
able to describe a thing yet not be able to imagine or 
make a mental image of it. There is an evident fallacy 
here. Either the description of four-space is no de- 
scription at all or it is a true delineation of an idealized 
construction which is well within the mind's powers of 
imagination. Indeed the question of imaginability is 
not determinative in itself; for what the mind may now 
be unable to imagine, because of its more or less 
nebulous character, and owing to its infancy may in 
the course of time be easily accomplished. 

The universe is a compacted plenum. It is chock- 
full of mind, of life, of energy and matter. These 
four are basically one. They exist, of course, in vary- 
ing degrees of tenuity and intensity and answer to a 
wide range of vibrations. Together, in their mani- 
festation of action and interaction, in their dynamic 
appearance, if you please, they constitute space. If these 
were removed with all that their existence implies there 
would result a condition of spacelessness in which no 
one of the appearances which we now perceive would 
be possible. Even sheer extensity would be non- 
existent. All scope of motility would be lacking. Di- 
mension, coordinates, direction, space-relations — all 
would be impossible. 

A straight line is an ideal construction of the mind. 
It does not exist in nature. It can never be actualized 
in the phenomenal universe. Between the ideal and 
the real, or actual, there is a kosmic chasm. It 
broadens or narrows according as the phenomenal ap- 
pearance approaches or recedes from the ideal. What, 
therefore, can be postulated of the one will not apply 
with equal force to the other. They are not congru- 



108 THE MYSTERY OF SPACE 

ent and can never be in the actualized universe. The 
moment the actual becomes identified with the ideal 
it ceases to be the actual. The universe does not exist 
as pure form, neither does space. As purely formal 
constructions of the intellect these can have no per- 
ceptible existence. The phenomenal or sensible may 
not be judged by exactly the same standard as the 
formal. The phenomenal or sensible represents things 
as they appear to the senses, or, so far as the actualized 
universe is concerned, as they really are. The formal 
represents things as they are made to appear by the 
mind. It cannot be actualized. It may be said that 
the purely formal is the limit of evolution. The phe- 
nomenal may approach the ideal as a limit, but can 
never become fully congruent with it. The difference 
between the ideal and the actual is a dynamic one; 
it is by virtue of this difference that the universe is 
held in manifestation. Evolution is the decrement of 
this difference between the purely formal and the 
actual. So long then as a kosmic differential is main- 
tained the phenomenal continues to be manifest : when 
it is finally reduced to nothing it goes out of mani- 
festation. The phenomenal is finite; the ideal infinite. 

Wherefore, it is undoubtedly improper to refer to 
space as being infinite. The term really is inap- 
plicable. Transfinity is much better and more accurate. 
Space is transfinite because its scope is greater than 
any finite scope of motility can encompass, because it 
exceeds finite comprehensibility. 

Riemann's notion that space is limited gains 
weight in the light of the foregoing considerations. 
But he could not conceive of the limitability and un- 
boundedness of space as such in its pure essence; but 



DIMENSIONALITY 109 

was compelled, by his own limitations, to make an 
idealized construction in which he could actualize 
his conception. And for real, dynamic space, he sub- 
stituted his ideal construction and proceeded upon that 
basis. And of course, his view while it had no refer- 
ence to perceptual space nevertheless possessed an 
illative relation thereto and should be recognized as 
construable in that light. 

The process of squaring the circle recognized as a 
geometric impossibility is significant of the fluxional 
nature of the universal residuum perpetually maintained 
between the archetypal and the manifested kosmos. It 
seems that there is a profound truth embodied in this 
problem. There is a lesson that may be learned by 
mathematicians, philosophers, scientists and thinkers in 
general. There is an element of eternal necessity and 
universality about it which is truly symbolic of the 
finity of the universe and the infinity of the archetypal. 
Just as a square or a series of polygonal figures in- 
scribed in a circle cannot be made to coincide exactly 
with the circle so cannot the actual be made to coin- 
cide with the ideal. The circumference of the circle is 
the unapproachable limit of inscribed squares. If it 
were possible so to multiply squares thus inscribed 
that a figure coincident with the circumference of a 
circle might be constructed, such a figure would not be 
a square but a circle. The manifested universe is like 
that — the process of inscribing squares within a circle. 
It is ever becoming, evolving, developing, but never 
quite attains. Infinity is a process. But no single 
stage in that process is infinite. Each is finite and their 
totality makes the infinity of the process. The universe 
manifested to the senses or the intellect is finite. 



no THE MYSTERY OF SPACE 

"Space," says Paul Carus, "is the possibility 
of motion in all directions." 4 To be sure, it is ad- 
mitted that space offers opportunity for motion in all 
directions. But is space this opportunity of motility? 
Or is possibility of motion space? The possibility of 
motion must rest in the thing that moves. It implies 
a potency in the moving entity, not in space. If it is 
meant that space is the potency that resides in the mov- 
ing element it is still more difficult to understand the 
connotation. But even granting this view, are we 
not compelled to recognize the dynamism of space 
as a necessary inference? Another definition which 
Carus gives is that space is a "pure form of exten- 
sion." If it be granted that space is a pure form of 
extension we should have to conclude that it has no 
Actual existence; for pure form does not exist except 
as an idealized construction. It cannot be found in 
nature. Pure form is ideal. Impure or natural form 
is actual. Therefore the space in which we live and 
in which the universe exists cannot be a "pure form" 
because life cannot exist in the purely formal. It is 
useless to talk about space as mere form so long as 
it maintains life. The difficulty which this phase of the 
question presents is another evidence of the inade- 
quacy of our definitions. 

It is also found to be impossible to concur in 
Carus' conception of knowledge a priori. His notion 
of the a priori varies somewhat from the Kantian view. 
He defines it as an "idealized construction," the "mind 
made," "abstract thought," and places it in the same 
category as a concept. This is undoubtedly born of 
his desire to get rid of Kant's "innate ideas" which 

*Vide Foundations of Mathematics, p. 107. 



DIMENSIONALITY in 

seem to be distasteful to him. But in doing so it 
appears that the real a priori has been overlooked. 
Let us examine for a moment this important question. 
The a posteriori connotates all knowledge gained 
through the senses, or sense experience. All knowl- 
edge therefore whose origin can be traced to the senses 
is knowledge a posteriori. Now, knowledge a priori 
should be just the opposite of this. It should indicate 
such knowledge as that which does not have its origin 
in the senses, or which is not dependent upon the ordi- 
nary avenues of sense-experience. Abstract thought is 
as truly experience as smelling, seeing or hearing. It 
is by traversing its scope of motility that the mind finds 
out what the norms of logic are. It could not remain 
quiescent and discover them. It has to be active, ex- 
amining, comparing and judging. Almost the entire 
range of thought, its entire scope, is characterized by 
the a posterioristic method. In fact, all thought is 
a posterioristic. Despite the fact that, in thinking in 
the abstract, it is necessary mentally to remove all ele- 
ments of concreteness, all materiality and all actuality, 
the conclusions reached have to be referred to the 
standards maintained by the actual, the concrete and 
the material. Then we do not start with the abstract 
in our thinking. We begin with the concrete and by 
mentally removing all physical qualities arrive at the 
abstract. 

The mind has a constitution. It acts in a given 
way because it is its nature so to act. Not be- 
cause it has learned to act in that manner. It per- 
forms certain functions intuitively without previous in- 
struction or experience for the same reason that water 
dampens or heat warms. It is natural for it to do so. 



ii2 THE MYSTERY OF SPACE 

This naturalness, this performance of function with- 
out being taught or without experience constitute the 
principle of apriority in the mind. Aprioriness is a 
principle of mind partaking of the very nature and 
essence of mind. It is the very mainspring of men- 
tality. Perception and conception are processes which 
the mind performs intuitively. The mind perceives 
and conceives because it is impossible for the normal 
mind to do otherwise. We take a view upon a given 
question; we assume certain mental attitudes of 
affirmation, negation or indifference because we have 
learned to do so by virtue of the tuitional capability 
of mind. These describe the a posteriori. That is, 
all knowledge obtained as a result of voluntary mental 
processes constitutes the mass of knowledge a pos- 
teriori. The a priori is what the mind is by nature: 
the a posteriori is what the mind becomes. It is the 
mind-content. 

The a priori is not a mental construction; it is an 
essential principle of mind. It should not be identi- 
fied with the "purely formal," as is done by Paul 
Carus : 5 

He says: 

"The a priori is identical with the purely formal 
which originates in our mind by abstraction. When 
we limit our attention to the purely relational, drop- 
ping all other features out of sight, we produce a 
field of abstraction in which we can construct purely 
formal combination, such as numbers, or the ideas 
of types and species. Thus we create a world of 
pure thought which has the advantage of being ap- 
plicable to any purely formal consideration and we 

• Vide Foundations of Mathematics, p. 42. 



DIMENSIONALITY 113 

work out systems of numbers which, when count- 
ing, we can use as standards of reference for our 
experience in practical life." 

Thus Carus definitely links up the a priori to a 
factor which is nothing more nor less than a mental 
by-product. For such is the category in which would 
be placed both the process of abstraction and its re- 
sults. It is therefore exceedingly difficult to under- 
stand why so cursory a consideration should have been 
given to the principle of apriority than which no other 
element of mind is more essentially a part of the mind 
itself. 

The formal is symbolic. It signifies an informing 
quantity. Pure form itself is but a negation of that 
which formerly filled it. Then, too, the formal is 
purely artificial because it is a mental construction. 
Essentially there is as much difference between the 
purely formal and the a priori as between creator and 
creature, as between potter and clay. The one is the 
builder, the other is the material; the one the knower 
and the other the known. Thus, the only reason that 
the formal is found to be answerable to the a priori 
at all is due to the fact that it is construable only upon 
the basis of the a priori. But being so is not sufficient 
warrant for its identification with the a priori. The 
formal merely represents the totality of possibilities 
in the universe as viewed by the mind; but as the 
number of possibilities open to the mind is, on account 
of its nature and purpose limited, it is not to be sup- 
posed that it (the mind) shall measure up to all the 
possibilities offered by the formal. Moreover, it is 
certain that no sane mind cherishes the hope that there 
shall ever be found in the universe of life and form a 



ii4 THE MYSTERY OF SPACE 

congruence for all of the possibilities held out by the 
purely formal. 

As an eternal principle of mind, the a priori is in 
agreement with the divine mind of the kosmos. In its 
aposteriority the mind is of diverse tendences, qualities 
and characteristics. Apriorily, it acts in unison with 
the eternal purpose of life and the universal mind. 
In its aposteriority, it often goes awry. In its apriority 
it can never be insane; insanity is a symptom of the 
morbid a posteriori. 

The mind in man acts the same as mind in the 
vegetal and lower animal kingdoms. Metabolism and 
katabolism, indeed all cell-activity, are a priori per- 
formances of the mind. Growth and all its phe- 
nomena, the cyclicism of natural processes, and every 
activity connected therewith belong to the category of 
the a priori. Cells multiply, divide, build up and tear 
down tissues and they do it intuitively. Most cer- 
tainly these functions are performed without any as- 
sistance from the intellect. All the myriad activities 
in nature with which the intellect in man has not the 
slightest concern, truly acting in accord with some 
primordial impetus, are activities a priori. 

Now what is the attitude of the intellect, in the 
light of the a priori, towards space and the question 
of dimensionality? It is evident that no matter what 
this attitude may be it is in agreement with the con- 
stitution of things and of the universe. And if so, it 
is right and without illusion. It is also evident that 
whatever notion a posteriori the intellect may enter- 
tain with respect to these questions is unavoidably 
liable to the illusionary drawbacks common to con- 
clusions based upon limited experience. The geometric 



DIMENSIONALITY 115 

view of space belongs to the category of the a pos- 
teriori. Hence it is subject to the usual imposition of 
error. 

Tersely stated, Kant's view of space is that it is 
a form of intuition, a form a priori, a transcendental 
form. As such he considered it to be a native form of 
perception not belonging to the category of sense- 
deliveries. Accordingly, space is a form of intuition 
arising out of and inhering in the constitution of mind. 
It is a notion which constitutes the universal and 
eternal prerequisite of mind and is, therefore, in- 
trinsically necessary to all phases of mentation. Now, 
this being true just what may be said to be the relation 
of dimensionality to this a priori form of space which 
is found to exist in the mind as an eternal aspect of 
its nature ? Does the mind intuitively measure its con- 
tents or its operations by the empirical standard of 
space-measurement known as dimension? Is the atti- 
tude of the mind towards the objectively real one of dis- 
crimination a priori as to the direction or dimension in 
which a percept may originate? In other words, does 
the mind habitually and intuitively refer its data to a 
system of coordinates for final determination? There 
is no other answer but that the mind makes no such 
reference and is dependent upon no kind of coordinate 
system in any of its operations a priori. As a form 
of intuition, the space notion is present in the mind 
as a scope of existence, of motility, of being and of 
sheer roominess. The notion of direction or dimen- 
sion, being an artificial construction, does not enter into 
this form of intuition at all. It is only when the mind 
comes to elaborate upon its perceptive performances 
and possibilities that the questions of relations, posi- 



n6 THE MYSTERY OF SPACE 

tions and directions arise. But this latter is a matter 
separate and distinct from the state of awareness 
which embodies the notion of space. 

Dimension is an arbitrary norm constructed by the 
mind for the determination of various positions in 
space. It is an accident or by-product of the process 
of elaborative cognition, a convenient and appropriate 
means of measurement for objects in space and their 
space-relations. But it is no more a priori than a foot 
rule or a square. But being purely an empirical product 
it may be said to be an aspect of psychogenesis because 
it relates to the evolutionary aspect of mind. The 
assumption may therefore be allowed that the mind 
may, in the course of its evolution, find it convenient 
and appropriate to devise an additional ordinate or 
dimension to satisfy the necessities of its more com- 
plex ramifications into the nature of things and to de- 
termine their greatly increased space-relations. It may 
be even possible for the mind to function normally in 
a space of four dimensions. But this would simply 
be a new adjustment, not a change in the essential 
nature of mind. It would be like the series of adjust- 
ments to environments which man has made in the on- 
ward movement of civilization. There has been no 
serious change in the manhood per se of man. That 
has remained the same; there has been merely a com- 
plication of environmental influences. Similarly, in the 
acquisition of four-dimensional powers, granting that 
such an acquisition is possible, there is nothing to be 
added to the aprioriness of mind itself. Is it not, 
therefore, logical to assume that the discovery of a 
fourth coordinate and the consequent conceptualiza- 
tion of the same, point to the development in the mind 



DIMENSIONALITY 117 

of a greatly extended faculty, more keenly penetrative 
powers of cognition and a further diversification of its 
environments than it has hitherto enjoyed? Indeed, 
it seems so. 



CHAPTER V 

The Fourth Dimension 

The Ideal and the Representative Nature of Objects in the 
Sensible World — The Psychic Fluxional the Basis of 
Mental Differences — Natural and Artificial Symbols — 
Use of Analogies to Prove the Existence of a Fourth 
Dimension — The Generation of a Hypercube or Tesser- 
act — Possibilities in the World of the Fourth Dimension — 
Some Logical Difficulties Inhering in the Four-Space Con- 
ception — The Fallacy of the Plane-Rotation Hypothesis — 
C. H. Hinton and Major Ellis on the Fourth Dimension. 

The world of mathesis is truly a marvelous do- 
main. Vast are its possibilities and vaster still its 
sweep of conceivability. It is the kingdom of the 
mind where, in regal freedom, it may perform feats 
which it is impossible to actualize in the phenomenal 
universe. In fact, there is no necessity to consider the 
limitations imposed by the actualities of the sensuous 
world. Logic is the architect of this region, and for 
it there is no limit to the admissibility of hypotheses. 
These may be multiplied at will, and legitimately so. 
The chief error lies in the attempt to make them 
appear as actual facts of the physical world. 

Mathematicians, speculating upon the possibilities 
of mathetic constructions and forgetting the necessary 
distinctions which should be recognized as differen- 
tiating the two worlds, in their enthusiasm have been 

118 



THE FOURTH DIMENSION 119 

led into the error of postulating as qualities of the 
phenomenal world the characteristics of the conceptual. 
Accordingly, a great deal of confusion as to the proper 
limits and restrictions of these conceptions has arisen 
and there still may be found those who are enthusi- 
astically endeavoring to push the actualities of the 
physical over into the conceptual. But in assuming 
any attitude towards mathetic propositions, especially 
with a view to demonstrating their actuality, very care- 
ful discrimination as to the essential qualities and their 
connotations should be made. Hence, before taking 
up a brief study of the fourth dimension proper, it is 
deemed fitting to indicate some of the fundamental dis- 
tinctions which every student of these questions should 
be able to make with reference to the data which he 
meets. 

All objects of the sensible world have both an es- 
sential or ideal nature and a representative or sensuous 
nature. That is, they may be studied from the stand- 
point of the ideal as well as the sensuous. The repre- 
sentative nature is that which we recognize as the mode 
of appearance to our senses which, as Kant held, is 
not the essential or ideal character of the thing itself. 
For there is quite as much difference between the sensu- 
ous percept and the real thing itself as between an 
object and its shadow. In fact, a concept viewed in 
this light, may be seen to have all the characteristics 
of an ordinary shadow; for instance, the shadow of a 
tree. View it as the sun is rising; it will then be 
seen to appear very much elongated, becoming less in 
length and more distinct in outline as the sun rises to 
a position directly overhead. The elongation may 
again be seen when the sun is setting. Throughout the 



120 THE MYSTERY OF SPACE 

day as the sun assumes different angles with reference 
to the tree the proportions and definiteness of the 
shadow vary accordingly. Thus the angularity of the 
sun, the intensity and fullness of the light and the shape 
and size of the tree operate to determine the charac- 
ter of the shadow. 

Much the same thing is true of a sensuous repre- 
sentation. If we examine carefully our ideas of geo- 
metric quantities and magnitudes, it will be found that 
the concepts themselves are not identical with the 
objects of the physical world, but mere mental shadows 
of them. The angularity of consciousness, or the dis- 
tinctness of one's state of awareness, being analogous 
to similar attitudes in the solar influence are the main 
determinants of the character of the mental shadow 
or concept. Wherefore mathematical "spaces" or 
magnitudes are not sensuous things and have there- 
fore no more real existence than a shadow, and 
strictly speaking not as much; for a shadow may be 
seen, while such magnitudes can only be conceived. It 
may be urged that since we can conceive of such things 
they must have existence of some kind. And so they 
have, but it is an existence of a different kind from that 
which we recognize as belonging to things in the sensi- 
ble world. They have a conceptual existence, but not 
a sensuous one. Therein lies the great difference. 

To be sure, a shadow is a more or less true rep- 
resentation of the thing to which it pertains. That 
this is true can be established empirically. Similarly, 
the degree of congruity between objects and concepts 
likewise may be determined. If this were not true 
we should be very much disappointed with what we 
find in the phenomenal world and could never be 



THE FOURTH DIMENSION 121 

quite sure that the mentograph existing in our minds 
was a faithful representation of the thing which we 
might be examining. But really the foundation for 
such a disappointment is present in every concept, every 
percept with which the mind deals. This disappoint- 
ment, although in actual experience is reduced to an 
almost negligible quantity, is due to the failure of 
sensuous objects to conform wholly to the specific de- 
tails of the mental shadow or mentograph. This lack 
of congruence between the mental picture and the ob- 
ject itself is necessary for obvious reasons. It is 
markedly observable in the early efforts of a child in 
learning distances, weights, resistances, temperatures 
and the like. No inconsiderable time is required for 
the child to be able correctly to harmonize his sense- 
deliveries with actual conditions. Otherwise, the 
child would never make any of the ludicrous mistakes 
of judgment of which it is guilty when trying to get its 
bearings in the world of the senses. In the course of 
time the child gradually learns by experience that cer- 
tain things are true of objects, distances, temperatures, 
resistances, etc., and that certain things are not true 
of them. He learns these things by actually contact- 
ing various objects. He is then competent to render 
correct judgments, within certain limits, as to the con- 
ditions which he finds in the sensible world. And the 
allowances, equations and corrections which his motor, 
sensory and psychic mechanisms learn to make in child- 
hood serve for all subsequent time. And this is im- 
portant to remember; for the mature mind is apt to 
forget or overlook the adaptations which the child- 
mind has made in its growth. 

If there were no such differences between the con- 



122 THE MYSTERY OF SPACE 

cept and the thing itself, actual physical contact would 
not be necessary. For one could rely wholly upon the 
sense-deliveries and each sense might operate entirely 
independently of all the others as there would be no 
necessity to correct the delivery of one by those of the 
others. This, of course, raises the question as to the 
necessity of sense-experience at all under conditions 
where there would be no disparity between the thing 
itself and the ideal representation of it in the mind. 
The absence of this variable quantity would open to 
the mind the possibility of really knowing the essen- 
tial nature of objects in the phenomenal world, a con- 
dition of affairs which is admittedly now without the 
range of the powers of the mind. 

At any rate, the essential "thingness" of objects 
can never be comprehended by the mind until the 
diminution of this disparity between the object of sense 
and the mental picture of it w T hich exists in the con- 
sciousness has proceeded to such a limit as either com- 
pletely to have obliterated it or to such an extent 
that the psychic fluxion is so slight as not to matter. 

It is believed that the results of mental evolution, 
as the mind approaches the transfinite as a limit, will 
operate to minimize the fluxional quantity which sub- 
sists between all objects of sense and their ideal repre- 
sentation as data of consciousness. The conclusion 
that the mind of early men who lived hundreds of 
thousands and perhaps millions of years ago on this 
planet consumed a much longer time in learning the 
adjustments between the objects which it contacted in 
the sensuous world and the elementary representations 
which were registered in its youthful consciousness 
than is to-day required for similar processes seems to 



THE FOURTH DIMENSION 123 

be demanded, and substantiated as well, by what is 
known of the phyletic development of the mind in the 
human race, 

In view of the above, it is thought that the dura- 
tion of such simple mental processes served not only to 
prolong the physical life of the man of those early 
days, but may also account for the puerility and in- 
capacity of the mind at that stage. Not that the slow 
mental processes were active causative agencies in 
lengthening the life of man, but that they together 
with the crass physicality of man necessitated a longer 
physical life. This, perhaps in a larger sense than 
any other consideration, accounts for the fundamental 
discrepancies in the mind of the primitive man in com- 
parison with the efficiency of the mind of the present- 
day man. In view of the potential character of mind 
and in the light of the well graduated scale of its ac- 
complishments, it is undoubtedly safe to conclude that 
the quality of mental capacities is proportional to the 
psychic fluxional which may exist at any time between 
the ideal and the essential or real. Mental differences 
and potentialities in general may be due to the magni- 
tude of the psychic fluxional or differential that exists 
between the conceptual and the perceptual universe. 
In some minds it may be greater than in others. The 
chasm between things-in-themselves and the mental 
notion pertaining thereto may vary in a direct ratio to 
the individual mind's place in psychogenesis, and there- 
fore, be the key to all mental differences in this respect. 

Most certain it is that there may be marked fluctua- 
tions in the judicial approach of minds towards 
any psychic end. In other words, there is not only a 
fluxional or differential between the object and its rep- 



124 THE MYSTERY OF SPACE 

reservation, but also a differential between the ap- 
proach of one mind and another in the judicial de- 
termination of notions concerning ideas. In this way, 
differences of opinions as to the right and wrong of 
judgments arise. Indeed, there seem to be zones 
of affinity for minds of similar characteristics, or 
minds that have the same degree of differential; so 
that, in choosing among the many possible judg- 
ments predicable upon a species of data, all those 
minds having the same degree of psychic differen- 
tial discover a special affinity or agreement among 
themselves. Hence, we have cults, schools of thought, 
and various other sectional bodies that find a basis of 
agreement for their operations in this way. The out- 
come of this remarkable intellectual phenomenon is 
that there are as many different kinds of judgments as 
there are zones of affinity among minds. Various sys- 
tems of philosophy owe their existence to these con- 
siderations, and the considerations themselves flow 
from the fact that all intellectual operations are essen- 
tially superficial; because there is no means by which 
they may penetrate to the steady flowing stream of 
reality which pervades and sustains objects in the sensi- 
ble world. 

In view, therefore, of the foregoing and with 
special reference to geometric constructions, it is neces- 
sary in approaching a study of the four-space that it be 
understood at the outset that the fourth dimension can 
neither be actualized nor made objectively possible 
even in the slightest degree in the perceptual world; 
because it belongs to the world of pure thought and 
exists there as an "extra personal affair," separate and 
distinct from the world of the senses. 



THE FOURTH DIMENSION 125 

As says Simon Newcomb r 1 

"The experience of the race and all the refine- 
ments of modern science may be regarded as show- 
ing quite conclusively that, within the limits of our 
experience, there is no motion of material masses, in 
the direction of a fourth dimension, no physical 
agency which we can assume to have its origin in 
regions to which matter cannot move, when it has 
three degrees of freedom." 

There is, however, no logical objection to the study 
of the fourth dimension as a purely hypothetical ques- 
tion, if by pursuit of the same an improvement of 
methods of research and of the outlook upon the field 
of the actual may be gained. Hence, it is with this 
attitude of mind that we approach the consideration of 
the fourth dimension. 

Various efforts have been made to render the con- 
ception of a fourth dimension of space thinkable. The 
student of space has reasoned: u We say that there 
are three dimensions of space. Why should we stop 
here? May there not be spaces of four dimensions and 
more?" Or he has said: "If 'A* may represent the 
side of a square, A 2 its area, and A 3 the volume of a 
cube with edge equal to A ; what may A 4 , A 5 or A wth rep- 
resent in our space? The conclusion, with respect to 
the quantity A 4 , has been that it should represent a 
space of four dimensions. 

Algebraic quantities, however, represent neither 
objects in space nor space qualities except in a purely 
conventional manner. All efforts to justify the objec- 
tive existence of a fourth dimension based upon such 
^ide Science, Vol. VII, p. 2, No. 158, 1898. 



126 THE MYSTERY OF SPACE 

reasoning will, therefore, fail; because the basis of 
such arguments is itself faulty. In the sentence : "The 
man loves his bottle," the thing meant is not the bottle, 
but what the bottle contains. For the purpose of the 
figure the bottle signifies its contents. There is no 
more real connection between the bottle and what it 
contains than between any word and the object for 
which it stands. Words are said to be symbols of 
ideas. But they are not natural symbols; they are 
conventional symbols, made for the purpose of cata- 
loguing, indexing and systematizing our knowledge. 
Words can be divorced from ideas and objects, or 
rather have never had any real connection with them. 
There are two classes of natural symbols, namely; 
objects and ideas. These, objects and ideas, symbolize 
realities. Realities are imperceptible and incompre- 
hensible to the intellect which has aptitude only for 
a slight comprehension of the symbols of realities. 
For instance, a tree is a natural symbol. It represents 
an actuality which is imperceptible to the intellect. The 
intellect can deal only with the sensible symbol. It is 
a natural symbol ; because it is possible directly to trace 
a living connection between the tree and the tree- 
reality. That is, it would be possible so to trace out 
the vital connection between the tree and its reality 
if the intellect had aptitude for such tracery. But, in 
reality, since it has no such aptitude, it remains for the 
work of that higher faculty than the intellect which 
recognizes both the connection and the intellect's in- 
ability to trace it. Further, an object is called a 
natural symbol because it is the bridge between sensu- 
ous representation and reality. It is as if one could 
begin at the surface of an object and by a subtle proc- 



THE FOURTH DIMENSION 127 

ess of elimination and excortication arrive at the heart 
of the universum of reality. No such consummation 
may be reached by dealing with words which have 
merely an artificial relationship with the objects which 
they signify. Again, ideas, that is, ideas that are uni- 
versal in application and have their roots in the great 
ocean of reality, are natural symbols; because if it were 
possible to handle an idea with the physical hands 
it would be possible to arrive at the heart of that 
which it symbolized without ever losing our connec- 
tion with the idea itself. In other words, ideas and 
objects, unlike words, can never be divorced from that 
which they symbolize. Both, being of the same class, 
are the opposite poles of realities. This then is the 
difference between natural symbols and artificial sym- 
bols — that a natural symbol, such as objects and ideas, 
is copolar with reality whereas an artificial symbol, 
such as words, geometric constructions and the like 
not only lacks this copolarity but is itself a symbol 
of natural symbols. 

It is, therefore, inconceivable that because the 
algebraic quantity A 3 has been arbitrarily decreed to 
be a representation of the volume of a cube, every 
such quantity in the algebraic series shall actually 
represent some object or set of objects in the physical 
world. Even if it be granted that such may be the 
case, is it not certain that there is a limit to things 
in the objective universe? Yet there may not be any 
limit to algebraic or mathematical determinations. 
The material universe is limited and conditioned; the 
world of mathesis is unlimited and unconditioned save 
by its own limitations and conditions. It is irrational 



128 THE MYSTERY OF SPACE 

to expect that physical phenomena shall justify all 
mathematical predicates. 

There is perhaps no single mathematical desider- 
atum or consideration which may be said to be the 
natural symbolism of realities; for the whole of mathe- 
matical conclusions is a mass of artificial and arbitrary 
but concordant symbols of the crasser or nether pole 
of the antipodes of realism. It is exceedingly danger- 
ous, therefore, to predicate upon such a far-fetched 
symbolism as mathematics furnishes anything purport- 
ing to deal with ultimate realities. And those who 
insist upon doing so are either blind themselves to 
these limitations or are madly endeavoring to befog 
the minds of others who are dependent upon them 
for leadership in questions of mathematical import. 

Analogies have been unsparingly used in efforts to 
popularize the four-space conception and much of the 
violence which has been done to the notion is due to 
this vagary. The mathematical publicist, in trying to 
give a mental picture of the fourth dimension, examines 
the appearances of three dimensional beings as they 
might appear to a two dimensional being or duodim. 
He imagines a race of beings endowed with all the 
human faculties except that they live in a land of but 
two dimensions — length and breadth. He thinks of 
them as shadows of three dimensional beings to whom 
there are no such conceptions as "up" and "down." 
They can see nothing nor sense anything in any way 
that is without their plane. They can move in any 
direction within the plane in which they live, but can 
have no idea of any movement that might carry them 
without that plane. A house for such beings might 
be simply a series of rectangles. One of them might 



THE FOURTH DIMENSION 129 

be as safe behind a line as a tridim or three dimen- 
sional being would be behind a stone wall. A bank 
safe for the unodim would be a mere circle. A duodim 
in any two dimensional prison might be rescued by 
a tridim without the opening of doors or the breaking 
of walls. An action of a tridim performed so as to 
contact their plane would be to them a miracle, abso- 
lutely unaccountable upon the basis of any known fact 
to the unodim or duodim. A tridim might go into 
a house where lived a family of duodims, appear and 
disappear without being detected or its ever being dis- 
covered how he accomplished such a marvelous feat. 
Our miracles, after the same fashion, are said to be 
the antics of some four dimensional being who has 
similar access to our three dimensional world and 
whose actions are similarly inexplicable to us. So the 
analogies have been multiplied. But the temptation 
to apply the consequences of such reasoning to actual 
three-space conditions has been so great that many 
have yielded to it and have consequently sought 
actually to explain physical phenomena upon the basis 
of the fourth dimension. 

The utilitarian side of the question of hyperspace 
has not been neglected either. And so, early in the 
development of the hypothesis and its various con- 
notations, the attention of investigators was turned to 
this aspect of the inquiry. Strange possibilities were 
revealed as a result. For instance, it was found that 
an expert fourth dimensional operator is possessed of 
extraordinary advantages over ordinary tridimen- 
sional beings. Operating from his mysterious hiding 
place in hyperspace, he could easily appear and dis- 
appear in so mysterious a manner that even the most 



130 THE MYSTERY OF SPACE 

strongly sealed chests of treasures would be easily and 
entirely at his disposal. No city police, Scotland Yard 
detective nor gendarme could have any terrors for 
him. Drs. Jekyll and Messrs. Hyde might abound 
everywhere without fear of detection. Objects as 
well as persons might be made to pass into or out of 
closed rooms "without penetrating the walls," thus 
making escape easy for the imprisoned. No tridimen- 
sional state, condition or system of arrangements, 
accordingly, would be safe from the ravages of evilly 
inclined four dimensional entities. Objects that now 
are limited to a point or line rotation could in the 
fourth dimension rotate about a plane and thus 
further increase the perplexities of our engineering 
and mechanical problems; four lines could be erected 
perpendicular to each other whereas in three space 
only three such lines can be erected; the right hand 
could be maneuvered into the fourth dimension and 
be recovered as a left hand; the mysteries of growth, 
decay and death would find a satisfactory explanation 
on the basis of the fourth dimensional hypothesis and 
many, if not all, of the perplexing problems of 
physiology, chemistry, physics, astronomy, anthro- 
pology and psychology would yield up their mysteries 
to the skill of the fourth dimensional operator. 
Marvelous possibilities these and much to be desired! 
But the most remarkable thing about these so-called 
possibilities is their impossibility. It is this kind of 
erratic reasoning that has brought the conception of a 
fourth dimension into general disrepute with the popu- 
lar mind. It is to be regretted, too, for the notion 
is a perfectly legitimate one in the domain of mathesis 
where it originated and rightly belongs. 



THE FOURTH DIMENSION 131 

It is not to be wondered at that metageometricians 
and others should at first surmise that, in the four- 
space, they had found the key to the deep mysteries 
of nature in all branches of inquiry. For so vast was 
the domain and so marvelous were the possibilities 
which the new movement revealed that it was to be 
expected that those who were privileged to get the 
first glimpses thereof would not be able to realize fully 
their significance. But the stound of their minds and 
the attendant magnification of the elements which they 
discovered were but incidents in the larger and more 
comprehensive process of adjustment to the great out- 
standing facts of psychogenesis which is only faintly 
foreshadowed in the so-called hyperdimensional. The 
whole scope of inquiry connected with hyperspace is 
not an end in itself. It is merely a means to an end. 
And that is the preparation of the human mind for 
the inborning of a new faculty and consequently more 
largely extended powers of cognition. Metageo- 
metrical discoveries are therefore the excrescences of 
a deeper, more significant world process of mental un- 
foldment. They belong to the matutinal phenomena 
incident to this new stage of mental evolution. All 
such investigations are but the preliminary exercises 
which give birth to new tendencies which are destined 
to flower forth into additional faculties and capacities. 
So that it is well that the evolutionary aspect of the 
question be not overlooked; for there is danger of this 
on account of the magnitude and kosmic importance 
of its scope of motility. 

A geometric line is said to be a space of one dimen- 
sion. A plane is a space of two dimensions and a 
cube, a space of three dimensions. In figure 7 below, 



132 



THE MYSTERY OF SPACE 



the line ab is said to be one dimensional; because only 
one coordinate is necessary to locate a point-position 
in it. The plane, abed, figure 8, is said to be two di- 
mensional because two coordinates, ab and db are re- 
quired to locate a point, as the point b. The cube 
abedefgh, figure 9, is said to be tridimensional, because, 
in order to locate the point b, for instance, it is neces- 
sary to have three coordinates, ab, be and gb. The 
tesseract is said to be four dimensional, because, in 
order to locate the point b, in the tesseract, it is neces- 
sary to have four coordinates, ab, be, bb r and h'b, fig- 
ure 10. 



Fig. 7. 




Fig. 8. 

It will be noted that in figures 8, 9 and 10, the 
element of perpendicularity enters as a necessary de- 
termination. In figure 8, the lines ab and bd are per- 
pendicular to each other. Similarly, in Fig. 10, lines 
ab, be, bb' and h'b are perpendicular to one another. 
That is, at their intersections, they make right angles. 
Similarly, figures representing any number of dimen- 
sions may be constructed. 



THE FOURTH DIMENSION 133 





Fig. 10. — The Tesseract. 



i 3 4 THE MYSTERY OF SPACE 

The line ab represents a one-space. An entity 
living in a one space is called a "iincdim." The plane, 
abed, represents a two-space, and entities living in such 
a space are called duodims. The cube, abedefgh, 
represents a three-space and entities inhabiting such 
a space are called tridims. Figure 10 represents a' 
four-space, and its inhabitants are called quartodims. 
Each of the above-mentioned spaces is said to have 
certain limitations peculiar to itself. 

The fourth dimension is said to lie in a direction 
at right angles to each of our three-space directions. 
This, of course, gives rise to the possibility of gen- 
erating a new kind of volume, the hypervolume. The 
hypercube or tesseract is described by moving the 
generating cube in the direction in which the fourth 
dimension extends. For instance, if the cube, Fig. 
9, were moved in a direction at right angles to each 
of its sides a distance equal to one of its sides, a figure 
of four dimensions, the tesseract, would result. 

The initial cube, abcc'e'fhh', when moved in a direc- 
tion at right angles to each of its faces, generates the 
hypercube, Fig. 10. The lines, ad ', bb' , cc' , dd', ee' , 
ff) 99 '> hh', are assumed to be perpendicular to the 
lines meeting at the points, a, b, c, d, e, f, g, h. Hence 
a'b' , b'd, dd', d'a', ef, fg, gg' , g'e, represent the final 
cube resulting from the hyperspace movement. Count- 
ing the number of cubes that compose the hypercube 
we find that there are eight. The generating cube, 
abcc'e'f'hh', and the final cube, a'b', b'd, dd' , d'a', ef, fg, 
99 '> 9' e > make two cubes; and each face generates a 
cube making eight in all. A tesseract, therefore, is a 
figure bounded by eight cubes. 



THE FOURTH DIMENSION 135 

To find the different elements of a tesseract, the 
following rules will apply: 

1. To find the number of lines: Multiply the num- 
ber of lines in the generating cube by two, and add a 
line for each point or corner in it. E.g., 2X12 
= 24 + 8 = 32. 

2. To find the number of planes, faces or squares: 
Multiply the number of planes in the generating cube 
by 2 and add a plane for each line in it. E.g., 2X6 
+ 12 = 24. 

3. To find the number of cubes in a hyper cube: 
Multiply the number of cubes in the generating cube, 
one, by two and add a cube for each plane in it. E.g., 
2X1+6 = 8. 

4. To find the number of points or corners: Mul- 
tiply the number of corners in the generating cube by 
2. E.g., 2X8=16. 

In a plane there may be three points each equally 
distant from one another. These may be joined, form- 
ing an equilateral triangle in which there are three 
vertices or points, three lines or sides and one sur- 
face. 

In three-space there may be four points each equi- 
distant from the others. At the vertices of a regular 
tetrahedron may be found such points. The tetra- 
hedron has four points, one at each vertex, 6 lines and 
4 equilateral triangles, as in Fig. 11. 

In four-space, we have 5 points each equidistant 
from all the rest, giving the hypertetrahedron. This 
four dimensional figure may be generated by moving 
the tetrahedron in the direction of the fourth dimen- 
sion, as in Fig. 12. If a plane be passed through each 



136 



THE MYSTERY OF SPACE 



of the six edges of the tetrahedron and the new vertex 
there will be six new planes or faces, making 10 in 
all, counting the original four. From the new vertex 
there is also a tetrahedron resting upon each base 
of the original tetrahedron so that there are five tetra- 
hedra in all. A hypertetrahedron is a four-dimen- 
sional figure consisting of five tetrakedra, ten faces, 
10 lines and 5 points. 





Fig. 11. — Tetrahedron. 



Fig. 12. — Hypertetrahedron. 



Paul Carus 2 suggests the use of mirrors so 
arranged that they give eight representations of a cube 
when placed at their point of intersection. He says: 



"If we build up three mirrors at right angles and 
place any object in the intersecting corner we shall 
see the object not once, but eight times. The body 
is reflected below and the object thus doubled is 
mirrored not only on both upright sides but in addi- 
tion in the corner beyond, appearing in either of the 
upright mirrors coincidingly in the same place. 
Thus the total multiplication of our tridimensional 
boundaries of a four dimensional complex is ren- 
dered eight-fold. 

"We must now bear in mind that this repre- 
sentation of a fourth dimension suffers from all the 

3 Vide Foundations of Mathematics, pp. 93-94- 



THE FOURTH DIMENSION 137 

faults of the analogous figure of a cube in two 
dimensional space. The several figures are not eight 
independent bodies but are mere boundaries and 
the four dimensional space is conditioned by their 
interrelation. It is that unrepresentable something 
which they inclose, or in other words, of which they 
are assumed to be boundaries. If we were four 
dimensional beings we could naturally and easily 
enter into the mirrored space and transfer tridi- 
mensional bodies or parts of them into those other 
objects reflected here in the mirrors representing the 
boundaries of the four dimensional object. While 
thus on the one hand the mirrored pictures would 
be as real as the original object, they would not 
take up the space of our three dimensions, and in 
this respect, our method of representing the fourth 
dimension by mirrors would be quite analogous to 
the cube pictured on a plane surface, for the space 
to which we (being limited to our tridimensional 
space-conception), would naturally relegate the 
seven additional mirrored images is unoccupied 
and if we should make the trial, we would find it 
empty. 1 ' 

The utility of such a representation as that which 
Carus outlines in the above is granted, i.e., so far as 
the purpose which it serves in giving a general idea 
of what a four-space object might be imagined to be 
like, but the illustration does not demonstrate the ex- 
istence of a fourth dimension. It only shows what 
might be if there were a four-space in which objects 
could exist and be examined. We, of course, have 
no right to assume that because it can be shown by 
analogous reasoning that certain characteristics of the 
fourth dimensional object can be represented in three- 
space the possible existence of such an object is thereby 



1 38 THE MYSTERY OF SPACE 

established. Not at all. For there is no imaginable 
condition of tridimensional mechanics in which an 
object may be said to have an objective existence 
similar to that represented by the mirrored cube. 

But there are discrepancies in this representation 
which well might be considered. They have virtually 
the force of invalidating somewhat the conception 
which the analogy is designed to illustrate. For in- 
stance, in the case of the mirrored object placed at 
the point of intersection of the three mirrors built up 
at right angles to each other. Upon examination of 
such a construction it is found that the reflection of 
the object in the mirrors has not any perceptible con- 
nection with the object itself. And this, too, despite 
the fact that they are regarded as boundaries of the 
hypercube; especially is this true when it is noted that 
these reflections are called upon to play the part of 
real, palpable boundaries. If a fourth dimensional 
object were really like the mirror-representation it 
would be open to serious objections from all view- 
points. The replacement of any of the boundaries 
required in the analogy would necessarily mean the 
replacement of the hypercube itself. In other words, 
if the real cube be removed from its position at the 
intersection of the mirrors no reflection will be seen, 
and hence no boundaries and no hypercube. The 
analogy while admittedly possessing some slight value 
in the direction meant, is nevertheless valueless so far 
as a detailed representation is concerned. So the 
analogy falls down; but once again is the question 
raised as to whether the so-called fourth dimension can 
be established or proven at all upon purely mathe- 
matical grounds. It also emphasizes the necessity for 



THE FOURTH DIMENSION 139 

a clearer conception of the meaning of dimension and 
space. 

The logical difficulties which beset the hyperspace 
conception are dwelt upon at length by James H. 
Hyslop. He says: 3 

"The supposition that there are three dimen- 
sions instead of one, or that there are only three 
dimensions is purely arbitrary, though convenient 
for certain practical purposes. Here the supposi- 
tion expresses only differences of directions from an 
assumed point. Thus what would be said to lie 
in a plane in one relation would lie in the third 
dimension in another. There is nothing to deter- 
mine absolutely what is the first, second, or third 
dimension. If the plane horizontal to the sensorium 
be called plane dimension, the plane vertical to it 
will be called solid, or the third dimension, but a 
change of position will change the names of these 
dimensions without involving the slightest qualitative 
change or difference in meaning. 

"Moreover, we usually select three lines or 
planes terminating vertically at the same point, the 
lines connecting the three surfaces of a cube with 
the same point, as the representative of what is 
meant by three dimensions, and reduce all other 
lines and planes to these. But interesting facts are 
observable here. 1. If the vertical relation between 
two lines be necessary for defining a dimension, then 
all other lines than the specified ones are either not 
in any dimension at all, or they are outside the 
three given dimensions. This is denied by all 
parties, which only shows that a vertical relation 
to other lines is not necessary to the determination 
of a dimension. 2. If lines outside the three ver- 
tically intersecting lines still lie in dimension or 

"Vide Philosophical Review, Vol. V, 1896, p. 352, et. seq. 



140 THE MYSTERY OF SPACE 

are reducible to the other dimensions they may lie 
in more than one dimension at the same time 
which after all is a fact. This only shows that 
qualitatively all three dimensions are the same 
and that any line outside of another can only repre- 
sent a dimension in the sense of direction from a 
given point or line, and we are entitled to assume 
as many dimensions as we please, all within three 
dimensions. 

''This mode of treatment shows the source of 
the illusion about the 'fourth dimension.' The term 
in its generic import denotes commensurable quality 
and denotes only one such quality, so that the 
property supposed to determine non-Euclidean 
geometry must be qualitatively different from this, 
if its figures involve the necessary qualitative dif- 
ferentiation from Euclidean mathematics. But this 
would shut out the idea of 'dimension' as its basis 
which is contrary to the supposition. On the other 
hand, the term has a specific meaning which as 
different qualitatively from the generic includes a 
right to use the generic term to describe them 
differentially, but if used only quantitatively, that 
is, to express direction as it, in fact, does in these 
cases, involves the admission of the actual, not a 
supposititious, existence of a fourth dimension 
which again is contrary to the supposition of the 
non-Euclidean geometry. Stated briefly, dimension 
as commensurable quality makes the existence of 
the fourth dimension a transcendental problem, but 
as mere direction, an empirical problem. And the 
last conception satisfies all the requirements of the 
case because it conforms to the purely quantitative 
differences which exist between Euclidean and non- 
Euclidean geometry as the very language about 
'surfaces,' 'triangles,' etc., in spite of the prefix 
'pseudo,' necessarily implies." 



THE FOURTH DIMENSION 141 

Thus it would seem that those who have been 
most diligent in constructing the hyperspace concep- 
tion have been the least careful of the logical diffi- 
culties which beset the elaboration of their assumptions. 
Yet it sometimes requires the illogical, the absurd and 
the aberrant to bring us to a right conception of the 
truth, and when we come to a comparison of the two, 
truth and absurdity, we are the more surprised that 
error could have gained so great foothold in face of 
so overwhelming evidences to the contrary. 

The entire situation is, accordingly, aptly set forth 
by Hyslop when he says, continuing: 

"There are either a confusion of the abstract 
with the concrete or of quantitative with qualitative 
logic, ... so that all discussion about a fourth 
dimension is simply an extended mass of equivoca- 
tions turning upon the various meanings of the term 
dimension.' This when once discovered, either 
makes the controversy ridiculous or the claim for 
non-Euclidean properties a mere truism, but effectu- 
ally explodes the logical claims for a new dimen- 
sional quality of space as a piece of mere jugglery 
in which the juggler is as badly deceived as his 
spectators. It simply forces mathematics to tran- 
scend its own functions as defined by its own advo- 
cates and to assume the prerogatives of meta- 
physics." 

Shall we, therefore, assent to the imperialistic policy 
of mathematicians who would fain usurp the preserves 
of the metaphysician in order that they may exploit 
a superfoetated hypothesis? It is not believed that 
the harshness of Hyslop's judgment in this re- 
spect is undeserved. It is, however, regretted that the 



142 THE MYSTERY OF SPACE 

notions of mathematicians have been so inchoate as to 
justify this rather caustic, though appropriate criticism. 
For it does appear that the moment the mathematician 
deserts the province of his restricted sphere of motility 
and enters the realm of the transcendental, that moment 
he loses his way and becomes an inexperienced mariner 
on an uncharted sea. 

It is interesting to note that Cassius Jackson 
Keyser, 4 while recognizing the purely arbitrary char- 
acter of the so-called dimensionality of space, never- 
theless lends himself to the view that "if we think of 
the line as generating element we shall find that our 
space has four dimensions. That fact may be seen in 
various ways, as follows: 

"A line is determined by any two of its points. 
Every line pierces every plane. By joining the 
points of one plane to all the points of another, all 
the lines of space are obtained. To determine a 
line, it is, then, enough to determine two of its 
points, one in the one plane and one in the other. 
For each of these determinations two data, as 
before explained, are necessary and sufficient. The 
position of the line is thus seen to depend upon four 
independent variables, and the four dimensionality 
of our space in lines is obvious." 

Similarly he argues for the four dimensionality of 
space in spheres: 

"We may view our space as an assemblage of 
its spheres. To distinguish a sphere from all other 
spheres, we need to know four and but four inde- 
pendent facts about it, as say, three that shall de- 

4 Vide Monist, Vol. XVI, 1896, Mathematical Emancipations. 



THE FOURTH DIMENSION 143 

termine its center and one its size. Hence our space 
is four dimensional also in spheres. In circles, its 
dimensionality is six; in surfaces of second order 
(those that are pierced by a straight line in two 
points), nine; and so on ad infinitum." 

The view taken by Keyser is a typical one. 
It is the mathematical view and is characterized 
by a certain lack of restraint which is found to be 
peculiar to the whole scheme of thought relating to 
hyperspace. It is clear that the kind of space that 
will permit of such radical changes in its nature as 
to be at one time three dimensional, at another time 
four dimensional, then six, nine and even w-dimen- 
sional is not the kind of space in which the objective 
world is known to exist. Indeed, it is not the kind of 
space that really exists at all. In the first place, a line 
cannot generate perceptual space. Neither can a circle, 
nor a sphere nor any other geometrical construction. 
It is, therefore, not permissible, except mathematically, 
to view our space either as "an assemblage of its 
spheres, ,, its circles or its surfaces; for obviously per- 
ceptual space is not a geometrical construction even 
though the intellect naturally finds inhering in it a 
sort of latent geometrism which is kosmical. For there 
is a wide difference between that kosmic order which 
is space and the finely elaborated abstraction which the 
geometer deceives himself into identifying with space. 
There is absolutely neither perceptible nor imper- 
ceptible means by which perceptual space in anywise 
can be affected by an act of will, ideation or movement. 
Just why mathematicians persist in vagarizing upon 
the generability of space by movement of lines, circles, 
planes, etc., is confessedly not easily understood espe- 



144 THE MYSTERY OF SPACE 

cially when the natural outcome of such procedure is 
self-stultification. It is far better to recognize, as a 
guiding principle in all mathematical disquisitions re- 
specting the nature of space that the possibilities found 
to inhere in an idealized construction cannot be ob- 
jectified in kosmic, sensible space. The line of de- 
markation should be drawn once for all, and all 
metageometrical calculations and theories should be 
prefaced by the remark that: "if objective space were 
amenable to the peculiarities of an idealized construc- 
tion such and such a result would be possible," or 
words to that effect. This mode of procedure would 
serve to clarify many if not all of the hyperspace 
conceptions for the non-mathematician as well as for 
the metageometricians themselves, especially those 
who are unwilling to recognize the utter impossibility 
of their constructions as applied to perceptual space. 
We should then cease to have the spectacle of other- 
wise well-demeanored men committing the error of 
trying to realize abstractions or abstractionizing reali- 
ties. Herein is the crux of the whole matter, that 
mathematicians, rather than be content with realities 
as they find them in the kosmos, should seek to reduce 
them to abstractions, or, on the other hand, make their 
abstractions appear to be realities. 

Keyser proceeds to show how the concept of 
the generability of hyperspace may be conceived by 
beginning with the point, moving it in a direction 
without itself and generating a line; beginning 
with the line, treating it similarly, and generating 
a plane; taking the plane, moving it in a direction at 
right angles to itself and generating a cube; finally, 
using the cube as generating element and constructing 



THE FOURTH DIMENSION 145 

a four-space figure, the tesseract. Now, as a matter 
of fact, a point being intangible cannot be moved in 
any direction neither can a point-portion of sensible 
space be removed. Nevertheless, we quite agree with 
him when he asserts: 

"Certainly there is naught of absurdity in sup- 
posing that under suitable stimulation the human 
mind may, in the course of time, speedily develop 
a spatial intuition of four or more dimensions." 
(The italics in the above quotation are ours.) 

Here we have a tacit implication that the notion 
which geometers have heretofore designated as "di- 
mension" really is a matter of consciousness, of in- 
tuition, and therefore, determinable only by the lim- 
itations of consciousness and the deliveries of our 
intuitive cognitions. As a more detailed discussion of 
this phase of the subject shall be entered into when 
we come to a consideration of Chapter VI on "Con- 
sciousness as the Norm of Space Determinations" 
further comment is deferred until then. 

Now, as it appears certain that what geometers 
are accustomed to call "dimension" is both relative 
and interchangeable in meaning — the one becoming 
the other according as it is viewed — the conclusion 
very naturally follows that neither constructive nor 
symbolic geometry is based upon dimension as com- 
mensurable quality. The real basis of the non- 
Euclidean geometry is dimension as direction. For 
whatever else may be said of the fourth dimension 
so-called it is certainly unthinkable, even to the meta- 
geometricians, when it is absolved from direction al- 
though no specific direction can be assigned to it. It 



i 4 6 THE MYSTERY OF SPACE 

is agreed perhaps among all non-Euclidean publicists 
that the fourth dimension must lie in a "direction which 
is at right angles to all the three dimensions." But 
if they are asked how this direction may be ascertained 
or even imagined they are nonplused because they 
simply do not know. The difficulty in this connection 
seems to hinge about the question of identifying the 
conditions of the world of phantasy with those of the 
world of sense. There are distortions, ramifications, 
submersibles, duplex convolutions and other mathetic 
acrobatics which can be performed in the realm of the 
conceptual the execution of which could never be 
actualized in the objective world. Because these antics 
are possible in the premises of the mathematical 
imagination is scarce justification for the attempts at 
reproduction in an actualized and phenomenal uni- 
verse. 

One of the proudest boasts of the fourth dimen- 
sionist is that hyperspace offers the possibility of a 
new species of rotation, namely, rotation about a 
plane. He refers to the fact that in the so-called one- 
space, rotation can take place only about a point. For 
instance in Figure 7, the line ab represents a one-space 
in which rotation can take place only about one of 
the two points a and b. In Figure 8 which represents 
a two-space, rotation may take place about the line 
ab or the line cd> etc., or, in other words, the plane 
abed can be rotated on the axial line ab in the direc- 
tion of the third dimension. In tridimensional space 
only two kinds of rotation are possible, namely, rota- 
tion about a point and about a line. In the fourth 
dimension it is claimed that rotation can take place 
about a plane. For example, the cube in Figure 9, by 



THE FOURTH DIMENSION 



H7 



manipulation in the direction of the fourth dimension, 
can be made to rotate about the side abgf. 

A very ingenious argument is used to show how 
rotation about a plane is thinkable and possible in 
hyperspace. But with this, as with the entire fabric 
of hyperspace speculations, dependence is placed al- 
most entirely upon analogous and symbolic conceptions 
for evidence as to the consistency and rationality of 
the conclusions arrived at. 



"1 

I 
I 

I 
I 
I 
I 

I 

I 

I 
I 

• 
I 

1 



Fig. 13. 



It is urged that inasmuch as the rotation about 
the line be in Figure 13 would be incomprehensible 
or unimaginable to a plane being for the reason 
that such a rotation involves a movement of the 
plane into the third dimension, a dimension of which 
the plane being has no knowledge, in like manner 
rotation about a plane is also unimaginable or incom- 
prehensible to a tridim or a three dimensional being. 
It is shown, however, that the plane being, by making 
use of the possibilities of an "assumed" tridimension, 
could arrive at a rational explanation of line rotation. 



148 



THE MYSTERY OF SPACE 



Figure 14 offers an illustration by means of which 
a two dimensional mathematician could demonstrate 
the possibility of line rotation. He is already ac- 
quainted with rotation about a point; for it is the only 
possible rotation that is bservable in his two dimen- 
sional world. By conceiving of a line as an infinity 
or succession of points extending in the same direc- 




FiG. 14. 

tion; by imagining the movement of his plane in the 
direction of the third dimension thereby generating 
a cube and at the same time assuming that the lines 
thus generated were merely successions of points 
extending in the same direction, he could demonstrate 
that the entire cube Figure 14, could be rotated about 
the line BHX used as an axis. For upon this hypoth- 
esis it would be arguable that a cube is a succession 
of planes piled one upon the other and limited only 



THE FOURTH DIMENSION 149 

by the length of the cube which would be extending 
in the, to him, unknown direction of the third dimen- 
sion. He could very logically conclude that as a plane 
can rotate about a point, a succession of planes con- 
stituting a tridimensional cube, could also be con- 
ceived as rotating about a line which would be a 
succession of points under the condition of the hypoth- 
esis. His demonstration, therefore, that the cube, 
Figure 14, can be made to rotate around the line 
BHX would be thoroughly rational. He could thus 
prove line-rotation without even being able to actual- 
ize in his experience such a rotation. 

Analogously, it is sought by metageometricians to 
prove in like manner the possibility of rotation about 
a plane. Thus in Figure 16 is shown a cube which has 
been rotated about one of its faces and changed from 
its initial position to the position it would occupy when 
the rotation had been completed or its final position 
attained. 

The gist of the arguments put forward as a basis 
for plane-rotation is briefly stated thus: The face 
cefg is conceived as consisting of an infinity of lines. 
A cube, as in Figure 15, is imagined or assumed to 
be sected into an infinity of such lines, each line being 
the terminus of one of the planes which make up the 
cube. Each one of the constituting planes is thought 
of as rotating about its line-boundary which intersects 
the side of the cube. The process is continued in- 
definitely until the entire series of planes is rotated, 
one by one, around the series of lines which constitute 
the axial plane. Hence, in order that the cube, Figure 
16, may change from its initial position to its final 
position each one of the infinitesimal planes of which 



150 



THE MYSTERY OF SPACE 



the cube is assumed to be composed must be made to 
rotate about each one of the infinitesimal lines of 



»■ 




X 

r 




l 1 " M, I if 

I'll 11 '! 1 

I 'Ml I. 

III 1 'Ml, In 

II 1 MM 'I 

i'i M'1,1 ill! 

!i!!U! !l 

! ! ' i . , , i , i ; ' 1 1 1 


p 


fob* 




G 


''l'!!!! 1 ! "i I! 
jit'ii'h'ii'ij 

i^JM'jlMiJh 

' i ' ! ' ' 1 1 ' i ' ' I ! i 

l! i'i' iji' iii ! j 
1 1 1 II 1 1 J 1 } 1 1 1 J 1 


fi 


M»M| |!l l|l| 

^Lmjl ill 


-m 



Fig. 15. 




^tr 



Initial Position Final Position 

Fig. 16. — Plane Rotation 



which the plane used as an axis is composed. In this 
way, it is shown that the entire cube has been made 
to rotate about its face, cefg. This concisely, is the 



THE FOURTH DIMENSION 151 

"quod erat demonstrandum" of the metageometrician 
who sets out to prove rotation about a plane. Thus 
it is made to appear that in order that tridimensional 
beings may be enabled to conceive of four-space rota- 
tion, as in Figures 15 and 16, in which the rotation 
must also be thought of as taking place in the direc- 
tion of the fourth dimension, they must adopt the same 
tactics that a two dimensional being would use to 
understand some of the possibilities of the tridimen- 
sional world. 

It is, of course, unwise to assume that because a 
thing can be shown to be possible by analogical rea- 
soning its actuality is thereby established. This con- 
sideration cannot be too emphatically insisted upon; 
for many have been led into the error by relying too 
confidentially upon results based upon this line of 
argumentation. There is a vast difference between 
mentally doing what may be assumed to be possible, 
the hypothetical, and the doing of what is actually 
possible, the practical. 

In the first place, plane-rotation in the actual uni- 
verse is a structural impossibility. The very nature 
and constitution of material bodies will not admit of 
such contortion as that required by the rotation of 
a body, say a cube, about one of its faces. Let us 
examine some of the results of plane rotation. 1. The 
rotation must take place in the direction of the fourth 
dimension. Now, as it is utterly impossible for any 
one, whether layman or metageometrician, even to 
imagine or conceive, in any way that is practical, the 
direction of the fourth dimension it is also impossible 
for one to move or rotate a plane, surface, line or 
any other body in that direction. We are in the very 



i S 2 THE MYSTERY OF SPACE 

beginning of the process of plane-rotation so-called 
confronted with a physical impossibility. 2. Plane 
rotation necessarily involves the orbital diversion of 
every particle in the cube. This alone is sufficient to 
prohibit such a rotation; for it is obvious that the 
moment a particle or any series of particles is diverted 
from its established orbital path disruption of that 
portion of the cube must necessarily follow. This 
upon the assumption that the particles of matter are 
in motion and revolving in their corpuscular orbits. 
3. Plane-rotation necessitates a radical change in the 
absolute motion of each individual particle, electron, 
atom or molecule of matter in the cube and a conse- 
quent retardation or acceleration of this motion. This 
upon the hypothesis that the particles of matter are 
vibrating at the rate of absolute motion. 4. It pre- 
supposes a reconstitution of each atom, molecule or 
particle in the cube, changing the path of intra-corpus- 
cular rotation either from a right to left direction or 
from a left to right direction, as the case may be. 
The particles of matter in the cube will be acted upon 
in much the same manner as the particles in a glove 
when it is maneuvered in the fourth dimension. In 
describing this phenomenon, Manning says: 5 

"Every part by itself, in its own place is turned 
over with only a slight possible stretching and slight 
changing of positions of the different particles of 
matter which go to make up the glove." 

The slight stretching and slight changing of the 
positions of the particles referred to would be of small 

'Vide Fourth Dimension, Simply Explained, edited by H. P. 
Manning, p. 28. 



THE FOURTH DIMENSION 153 

consequence if applied to ponderable bodies. But 
when used in connection with particles of matter which 
are themselves of very infinitesimal size means far 
more — enough, as we have said, to militate severely 
against the integrity of the cube. It is not deemed 
necessary to go further into the physical aspects of 
plane-rotation as it is believed sufficient has been said 
to negative the assumption from a purely structural 
viewpoint. 

Among the vagaries of hyperspace publicists none 
is perhaps more notable than the view taken by C. H. 
Hinton : 6 

"If it could be shown that the electric current 
in the negative direction were exactly alike the elec- 
tric current in the positive direction, except for a 
reversal of the components of the motion in three 
dimensional space, then the dissimilarity of the dis- 
charge from the positive and negative poles would 
be an indication of the one-sidedness of our space. 
The only cause of difference in the two discharges 
would be due to a component in the fourth dimen- 
sion, which directed in one direction transverse to 
our space, met with a different resistance to that 
which it met when directed in the opposite direc- 
tion." 

To be sure. And with equal certainty it might 
be said that if the moon were made of green cheese 
it might well be the ambition of the world's chefs to 
be able at some time to flavor macaroni with it, thus 
serving a rare dish. Even so, if there were an actual, 
objective fourth dimension to our space we might be 
able to shove into it all the perplexing problems of 

'Vide Fourth Dimension, p. 75, C. H. Hinton. 



i 5 4 THE MYSTERY OF SPACE 

life and let it solve them for us. But the fact that 
the fourth dimensional hypothesis is itself a mere sup- 
position seems to have been overlooked or rather 
completely ignored by Hinton. Or else, ought 
it not be an obvious folly to hope to construct a ra- 
tional explanation of perplexing physical conditions 
upon the basis of a purely suppositionary, and there- 
fore unproven, hypothesis? 

The recognized domain of the four-space, mathe- 
matically considered, is according to the most generous 
allowance very small, so small, in fact, that the dis- 
position of some to crowd into it the essential content 
of the manifested universe is a matter of profound 
amazement. Then, too, it cannot be denied that there 
is no appreciable urgency or necessity for having re- 
course to a purely hypothetical construction for ex- 
plicatory data regarding a phenomenon which has not 
been shown to be without the scope of ordinary 
scientific methods of procedure to unravel. 

The claim of certain spiritualists, notably Zoll- 
NER of Leipsig, that the phenomena of spiritism 
is accountable for on the grounds that the fourth 
dimension affords a residential area for discar- 
nate beings whence spiritistic forayers may impose 
their presence upon unprotected three dimensional 
beings is no less fatuous than the original supposition 
itself. For upon this latter is built the entire fabric 
of meaningless speculations so gleefully indulged in 
by those who glibly proclaim the reality of the four- 
space. Indeed, clearer second thought will reveal that, 
when the pendulum of erratic thinking and trafficking 
in mental constructions swings back, hyperspaces, after 
all, are but the ignes fatuii of mathetic obscurantism. 



THE FOURTH DIMENSION 155 

Then, why should it be deemed necessary to dis- 
cover some more mysterious realm of four dimen- 
sional proportions in which the spirits of the dead may 
find a habitation? Are the spiritualists, too, reduced 
to the necessity of further mystifying their already 
adequately mysterious phenomena ? If there were not 
quite enough of physicality upon the basis of which 
all the antics of these entities can be explained, and 
that satisfactorily, one would, as a matter of course, 
be inclined to lend some credence to these claims; but 
as it is clear that all organized beings have some 
power, if no more than that which maintains their 
organization, and as it ought also be an acceptable fact 
that such a being is directed by mind; and further, 
that owing to the nature of a spirit body it can pene- 
trate solid matter or matter of any other degree of 
density below the coefficient of spirit matter, it ought 
likewise be unnecessary to go without the province 
of strictly tridimensional mechanics for an explanation 
of spiritistic phenomena. 

Equally unnecessary and uncalled for is the attempt 
of certain others who lean toward the view of specu- 
lative chemists to account for the none too securely 
established hypothesis that eight different alcohols, 
each having the formula C 5 H 12 may be produced 
without variation. This is said to be due to the fact 
that certain of the component atoms, notably the 
carbon atoms, take a fourth dimensional position in 
the compound and thus produce the unusual spectacle 
of eight alcohols from one formula. Have chemists 
actually exhausted all purely physical means of reach- 
ing an understanding of the carbon compounds and 
are therefore compelled to resort to questionable 



156 THE MYSTERY OF SPACE 

means in order to make additional progress in their 
field? It is incredible. Hence the more facetious 
appears the mathematical extravaganza in which 
originates the tendence among the more sanguine ad- 
vocates to make of the fourth dimension a sort of "jack 
of all trades," a veritable "Aladdin's lamp" wherewith 
all kosmic profundities may be illuminated and made 
plain. Not until the perfection of instruments of pre- 
cision has been reached, and not until human ingenuity 
has been exhausted in its efforts to produce more re- 
fined methods of research should it be permissible 
even to venture into untried and more or less deba- 
table fields in search of a relief which after all is 
unobtainable. 

Notwithstanding the fact that all attempts at ac- 
counting for physical phenomena on the basis of 
^dimensionality (which is itself by all the standards of 
objective reference a non-existent quantity and therefore 
irreconcilable with perceptual space requirements) are 
to be characterized simply as a senseless dalliance with 
otherwise deeply profound questions, many have fallen 
into a complete forgetfulness of the logical barriers 
inhering in and hedging about the query and have 
committed other and less excusable errors in the 
premises. Take, for instance, the suggestion that the 
action of a tartrate upon a beam of polarized light 
is due to the assumption of a fourth dimensional 
direction by some component in the acid. This for the 
reason that experimentation has shown that tartaric 
acid, in one form, will turn the plane of polarized 
light to the right while in another form will turn it 
to the left. It is not believed, however, that there is 
any warrant for such an assumption. There is also 



THE FOURTH DIMENSION 157 

another kind of tartrate which seems to be neutral in 
that it has no effect whatever upon the beam of light, 
turning it neither to the right nor to the left nor 
having other visible or determinable effect upon it. 
Indeed, it is not clear how it is hoped to prove such 
a case by constituting as a norm a hypothesis which 
is essentially indemonstrable. A more logical pro- 
cedure would be first to establish the objective, dis- 
coverable posture of four-space; show the actual 
movement of matter and entities therein; locate it by 
empirical methods of research, and then, basing our 
assertions upon apodeictic evidences, assume a new 
attitude toward these phenomena because of the sup- 
port found in established and verifiable facts. Some 
hope of gaining a respectful hearing might then be 
entertained; but at least to do so now appears to 
be quite untimely. 

Major Wilmot E. Ellis, Coast Artillery Corps, 
United States Army, in The Fourth Dimension Simply 
Explained, 1 remarks : 

". . . in the ether, if anywhere, we should ex- 
pect to find some fourth dimensional characteristics. 
Gravitation, electricity, magnetism and light are 
known to be due to stresses in, or motions of, the 
infinitesimal particles of the ether. The real nature 
of these phenomena has never been fully explained 
by three dimensional mathematical analysis. Indeed, 
the unexplained residuum would seem to indicate 
that so far we have merely been considering the 
three dimensional aspects of four dimensional 
processes. As one illustration of many, it has been 
shown both mathematically and experimentally that 

1 Q. v., p. 242, edited by H. P. Manning. 



1 58 THE MYSTERY OF SPACE 

no more than five corpuscles may have an inde- 
pendent grouping in an atom." 

The weakness of this view may be due to the fact 
that at that time Major Ellis was emphasizing in 
his own mind the necessity of simplifying the concep- 
tion so as to make it of easy comprehension rather 
than the establishment of any fealty to truth or the 
spirit of mathesis in his examination of the problem. 
What therefore of reality the student fails to find in 
his view may be attributed to the sacrifice which the 
writer (Major Ellis) felt himself called upon to 
make for the sake of simplicity. Hence a certain ex- 
pressed connivance at his position is allowable. But, 
on the other hand, if such were not the conscious 
intent of Major Ellis it is not understood how it 
should appear that "the unexplained residuum would 
seem to indicate that so far we have merely been 
considering the three dimensional aspects of four 
dimensional processes." Contrarily, it has yet to be 
proved that three dimensional space does not afford 
ample scope of motility for all observable or recog- 
nizable physical processes and that there is no neces- 
sity for reference to hyperspace phenomena for an 
explanation of the "unexplained residuum." It is, of 
course, understood that many of the possibilities 
predicated for hyperspace are purely nonsensical so 
far as their actual realization is concerned. Our con- 
cern is, therefore, not with that class of predicates, 
but with those wherein reside some slight show of 
probability of their response to the conditions of 
^-dimensionality either as a system of space-measure- 
ment or a so-called space or series of spaces. 



THE FOURTH DIMENSION 159 

Major Ellis concludes his simple study of four- 
space by proposing the following query: 

"May not birth be an unfolding through the 
ether into the symmetrical life-cell, and death, the 
reverse process of a folding-up into four dimen- 
sional unity?" 

It is confessed that there seems to be nothing to 
warrant the giving of an affirmative reply to this 
query. It is, perhaps, sentimentally speaking a very 
beautiful thing to contemplate death as a painless, un- 
conscious involvement into a glorious one-ness with all 
life, and birth, as the reverse of all this. But where 
is the utility of such a dream if it be merely a dream 
and impossible of realization? 

Simon Newcomb/ at one time one of the out- 
standing figures in the early development of the 
fourth dimensional hypothesis, openly declared that 
"there is no proof that the molecule may not vibrate 
in a fourth dimension. There are facts which seem 
to indicate at least the possibility of molecular motion 
or change of some sort not expressible in terms of 
time and the three coordinates in space." 

Of course, there is no proof that a molecule may 
not at times be ensconced in a four-space neither is 
there proof nor probability that it is so hidden. In- 
deed, there is no proof that there is such a thing as 
a molecule for that matter. 

In all of the foregoing proposals it is assumed 
that the fourth dimension really exists and that it lies 
just beneath the surface of the visible, palpable limits 

•Vide Science, Vol. VII, 158, 1898, p. 4. 



160 THE MYSTERY OF SPACE 

of the material universe; that lying in close juxtaposi- 
tion to all that we are able to see, to hear or sense 
in any way is this mysterious, eternally prolific, all- 
powerful something, hyperspace, ever-ready to nourish 
and sustain the forms which have the nether parts 
firmly encysted in one or the other of her w-dimen- 
sional berths. Thus it would seem that while yet 
functioning in a strictly tridimensional atmosphere, 
some one, more reckless than the rest, should at last 
stumble upon some up-lying portion of it and be in- 
stantly transformed into a mathetic fay of etherealized 
four-dimensional stuff. 



PART TWO , 
SPATIALITY 

AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE 
AS DISTINGUISHED FROM THE MATHE- 
MATICAL INTERPRETATION 



CHAPTER VI 

Consciousness t.he Norm of Space 
Determinations 

Realism Is Determined by Awareness — Succession of Degrees 
of Realism — Sufficiency of Tridimensionality — The In- 
sufficiency of Self-Consistency as a Norm of Truth — Gen- 
eral Forward Movement in the Evolution of Conscious- 
less Implied in the Hyperspace Concept — The Hypothet- 
cal Nature of Our Knowledge — Hyperspace the Svmbol 
of a More Extensive Realm of Awareness — Variations in 
the Method of Interpreting Intellectual Notions — The 
Tuitional and the Intuitional Faculties — The Illusionary 
Character of the Phenomenal — Consciousness and the De- 
grees of Realism. 

Things have value for us only to the extent to 
which we can become aware of their being. The 
appraisement of all objects, conditions, states or quali- 
ties is determined directly by the degree or quality 

161 



162 THE MYSTERY OF SPACE 

of awareness with which we apprehend them. Those 
elements which are without the intellect's scope of 
awareness have no interest and hence no value so far 
as the individual intellect is concerned. And this is 
true of all degrees and states of consciousness from 
the lowest to the highest, from the human to the 
divine. 

There enter into all conscious determinations three 
factors, namely: (a) the scope, or totality, of adapta- 
tions which an organism -can make in the sensible 
world, (b) the power of consciousness to make adapta- 
tions and (c) environment. These three are inter- 
dependent. The totality of adaptations depends 
primarily, of course, upon the quality of conscious 
powers or faculties, and also, in a lesser degree, upon 
opportunities afforded by environment. Faculties of 
consciousness are derived directly from the influences 
exerted upon the organism by his environment and 
the results of the struggle to overcome them. En- 
vironment is of two kinds, artificial and natural. The 
artificial environment is such as has been modified by 
our conscious action upon external phenomena. The 
residue is natural. And thus the scope of adaptability 
becomes an unvarying witness to the quality of con- 
sciousness manifesting through a given organism. 

The universe is so constructed that the essential 
character of its various states and qualities is a fixed 
quantity for a given scope of consciousness and varies 
only as the sphere of consciousness varies. States of 
existence or scopes of adaptation which are registering 
upon a higher plane or in a more subtle sphere of 
existence than that in which we may at any time be 
functioning can only appear evidential to us when the 



NORM OF SPACE DETERMINATIONS 163 

mechanism of our consciousness becomes congruently 
adjusted therewith. So that the focus of consciousness 
must always be a variable quantity adaptable, under 
proper conditions, to any plane in the kosmos. Con- 
sciousness, then, becomes the sphere of limits both 
of knowledge and adaptability. But lest we seem to 
admit implicitly part of the contentions which mathe- 
matical publicists have made in postulating the unodim 
and duodim consciousness, it is necessary carefully to 
differentiate between the results arrived at as a result 
of the two procedures. In the first place, analysts 
assume the existence of a unodim and duodim plane 
of consciousness and proceed to construct thereon an 
analogy designed to show the feasibility of another 
assumption, the fourth dimension. While, in laying 
the foundation of consciousness upon a tridimensional 
plane we do not start with an assumption, but with 
a fact. Therein lies the difference. Enormous ad- 
vantages inhere in a procedure based upon facts, but 
in a series of planes built upon assumptions no such 
advantages are discovered. For however much the 
series of hypothetical planes may be extended or 
elaborated there must inhere necessarily throughout 
the series an assumptional value which vitiates the 
conclusions no less than the premises. The sanity 
and integrity of intellectual operations depend almost 
entirely upon the differentiation which we make be- 
tween the necessities arising out of assumptions and 
those which spring up empirically from established 
facts. No procedure is necessary to establish the 
value of such a differentiation, nevertheless it may be 
suggested that it is allowable, under the rules of logic, 
to make any assumption whatsoever so long as care 



164 THE MYSTERY OF SPACE 

is taken to see that the conclusions embody in them- 
selves the characteristics of the original premise. 
For instance, it is permissible to assume that space 
is curved. Under such an assumption, it is only nec- 
essary that the constructions which follow shall be 
self-consistent. But the case is different when we come 
to deal with spatiality and vitality. These are quanti- 
ties which cannot, in the last analysis, be made to 
conform to the rules of the game of logic. 

Thus, when it is intimated that realism lends itself 
to an apparent division into degrees, and that each 
degree has a corresponding state of consciousness, it 
is by no means to be inferred that such apparent divi- 
sions are of mathematical import. For, in reality, 
i.e., when the consciousness has expanded so as to 
become congruent with the limits of even the space 
mind (vide Fig. 20), there appear to be no divisions 
in realism. It is only because of the fragmentariness 
of our outlook upon the kosmos that realism appears 
to be divided into various planes; for all of these 
planes are one. The divisions exist for relative knowl- 
edge, but not for complete knowledge; they exist for 
a finite intelligence, but not for a transfinite intelli- 
gence. That is why we view realism as a series of 
planes. It is because we discover that, as we proceed, 
as our consciousness expands and we take in more and 
more of the vital activities of the kosmos and under- 
stand better the causes underlying that which we 
contact, we have passed from a state of lesser knowl- 
edge to one of greater knowledge. And so we say 
we have passed from one degree of realism to an- 
other, whereas, really we have not passed from one 



NORM OF SPACE DETERMINATIONS 165 

degree of realism to another degree. Instead, it is 
our consciousness that has expanded. 

If now, we conceive reality to be a scale extending 
from one extremity to another (that is, from supreme 
consciousness to entire unconsciousness, from final 
knowledge to total ignorance), and the intellectual 
consciousness as the indicator which traverses the 
scale denoting at all times the precise degree of our 
comprehension of reality, and hence the degree of 
expansion of consciousness, we shall constitute a simili- 
tude closely approximating the real status quo of 
humanity with respect to the sensible and supersensible 
worlds. The quantity or force which causes the indi- 
cator to move along the scale is the quality of aware- 
ness. And this varies directly as the scope of adapta- 
bility varies. Realism is homogeneous throughout its 
extent; but the scale marked upon it registers from 
naught to unity. And between these every conceiv- 
able degree of awareness may be registered. The 
indicator moves only as the scope widens, and thus is 
shown a change in the quality of awareness. For, 
however paradoxical it may seem, the wider the scope 
of knowledge the better its quality: the more one 
knows, the more complete and of higher quality be- 
comes that which he knows. 

The intellect is of scientific tendence, studiously 
rejecting all phenomena which do not yield to its 
senso-mechanisms. Even intuitions suffer the humility 
of rejection and do not escape the limitations which 
the intellect imposes upon them. This is so, because, 
as yet, there is no adequate perceptive and conceptive 
apparatus for the propagation and classification of 
intuitions, as apart from concepts. The outcome of 



166 THE MYSTERY OF SPACE 

these proscriptions is that intuitions — free, mobile, and 
more or less formless in themselves, must first be 
rehabilitated and vestured in garments a la intellect 
to conform to the prevailing mode. But intuitions 
thus treated are no longer intuitions, but empirical 
concepts. True intuitions are like aqueous vapor — 
amorphous, permeating, diffusive: axioms or empirical 
concepts are like cakes of ice — formal, inflexible and 
conforming to the shape of the mold into which they 
are poured. Because of this — the scientific tendence 
of the intellect and the consequent necessity of re- 
forming so much of the data which constitute its sub- 
structure, of pressing, condensing and reshaping it to 
suit its own ready-made patterns — it can be perceived 
how profound is the influence of the intellectual con- 
sciousness in determining the character of the totality 
of data which the sensible world, and for that matter, 
the supersensible, offer us. The intellect is the only 
means at hand for the interpretation of the meaning 
and significance of the world of phenomena. Conse- 
quently, whatever meaning or significance we -are led 
to attach to that part of the universe which we con- 
tact, in any way, is dictated by the intellectual con- 
sciousness. There is no escape from the decisions of 
the intellect so long as the present scheme of things 
endures. 

Thus, by whatever standard of reference the 
matter may be determined, it remains indisputably 
established that the intellectual consciousness is the 
sole determinant of the phenomenal value of every- 
thing within our scope of awareness or adaptability. 
And whatever the fault, incongruity or discrepancy 
that may be revealed by a more intimate knowledge 



NORM OF SPACE DETERMINATIONS 167 

of the genesis and character of the appearance of 
the sensible world, it will be found to be due to the 
peculiar cut and mode of the intellect and not to 
things themselves. The value, qualitative or existen- 
tial, which the intellect irrevocably assigns to objects 
and conditions in the world of the senses is the ex- 
clusive norm not only by which these are judged, but 
also, by which our action upon them and their action 
upon us are determined. Images or objects which do 
not act upon us and upon which we cannot act have 
no interest for us. But as an integral part of the 
totality of images or objects in the sensible world, 
we must inevitably act upon all that is outside of 
ourselves, and these, in turn, must react upon us. On 
the other hand, there must be objects and images in 
the universe of life and form upon which, because of 
their inherent nature and on account of the lack of 
our interest in them and their interest in us, we can 
neither act nor become the object of their action. 

But herein is a mystery. For, either we act upon 
and are recipients of the action of the totality of 
images or objects in both the sensible and super- 
sensible worlds, or we are so placed in the grand 
scheme of things that both ourselves and the sphere 
of our interests and possible actions are closed cir- 
cuits, hermetically sealed and non-communicative with 
the other like spheres, which do not and cannot act 
upon us. There is yet a third possibility — that we are 
so fashioned, in the entirety of our being, that some 
part of us is exactly congruent with some part of 
every sphere of possible actions and interests in the 
kosmos, and therefore, each of us has being or con- 
sciousness of a kind that is keyed to and registering 



168 THE MYSTERY OF SPACE 

in the totality of such spheres; and that, at present, 
because our interests and possible actions are limited 
to the plane of sensibility, we are conscious only there. 
And further, that although those spheres of our con- 
sciousness which are fixed to register in other planes 
do not answer to the lowest on which we now operate, 
having a character of which we are unaware, they 
nevertheless cannot be said not to exist, because of 
the lack of communication between them. Among 
these three possible choices, we have no hesitancy in 
expressing a decided preference for the last mentioned 
— that the range of our being is co-extensive with 
the range of reality, and like a pendulum, we oscillate, 
at long intervals, between two kosmic extremities — 
nescience and omniscience. 

The intellectual consciousness is the touch-stone of 
realism. It is like a spreading light which, as it ex- 
pands, reveals previously darkened corners and con- 
ditions, only it has power both to reveal and to bring 
into manifestation. In its present state, man's con- 
sciousness is like a searchlight. It illumines and takes 
cognizance of everything that falls within its scope of 
motility and is consequently able to study in detail 
that which it reveals. But that which is beyond its 
scope is as if it never existed so far as the individual 
consciousness is concerned. It is not reasonable to 
predict that the same characteristics that are observ- 
able in any given state shall persist throughout all 
the various scopes through which the consciousness 
must proceed in its evolutionary expansion. For the 
scale of kosmic realism is one grand panorama ex- 
tending from the grossest to the most subtle and re- 
fined. While in general the thread of realism may 



NORM OF SPACE DETERMINATIONS 169 

pervade the entire scale it is nevertheless marked by 
many and diverse changes in its characteristics as it 
is followed from one stage to another. So that the 
realistic character of one stage may vary greatly from 
that which next preceded it or from that which will 
succeed it. It would appear, therefore, that in pass- 
ing from one stage of realism to another there need 
not remain anything but the mere fact of reality in 
its connection with ultimate reality; for it is obvious 
that in every condition of realism which may be en- 
countered in the kosmos there must be a basic thread 
of ultimate reality running through the whole. The 
entire gamut of realism may accordingly be traversed 
without the danger of being diverted from the golden 
thread of realism which thus permeates all. It is 
always the phenomena of realism with which we are 
concerned and which we are trying to understand 
rather than realism itself. It is this that confounds 
us. If it were not for the phenomena, which is the 
way realism or life presents itself to our consciousness, 
we should experience no trouble in discovering the 
reality, all other things being equal. For the former 
ever obscures the latter. It is the supreme task of 
mental evolution to break through the clouds of phe- 
nomena in the search for the eternal substratum of 
reality which runs through the sensible universe of 
things. 

The first view of conditions that the mind takes 
upon awakening to consciousness in any new sphere 
of cognition is necessarily hazy and inchoate. There 
is more or less of astonishment, wonder and bewilder- 
ment upon first becoming aware of a new scope of 
realism. In this state it is natural that the mind should 



170 THE MYSTERY OF SPACE 

overlook or ignore much that is essential and perhaps 
all that is so even escaping the true import of the 
phenomena which it senses. It is reasonable, too, that 
in such a state the main outlines of what is really 
seen may be greatly distorted and exaggerated so that 
it is well-nigh impossible to secure a correct compre- 
hension of the character of a new scope of realism 
from any early survey. It is not until later years, 
after much study and circumspection that the mind, 
becoming used to the new conditions, begins to get 
correct impressions and to make valid judgments as 
to that which it discerns. And even then, it not in- 
frequently happens that the resultant view of things 
in general is found to be in need of revision and cor- 
rection. Hence, after everything is sifted down to 
the ultimate allowance for the illusion incident to too 
much enthusiasm and wonder we have only a very 
small residuum of truth upon which to build and this 
latter we often find to be the single thread of reality 
which runs through all the phenomena and which is, 
therefore, the only quantity that remains worthy of 
much consideration. 

Thus it is with religion. The path of progress 
over which our religious conceptions have come need 
not be outlined here, but to any one at all acquainted 
with the history of religious thought and ideals it 
at once must be patent that it has been one continuous 
surrender of the old for the new, of one degree of 
realism for another newer and higher degree; that 
always it has been the phenomena, the flora of the 
ideals which have had to give way, while nothing was 
left but the roots of realism from which they have 
sprung. It has been the same with scientific knowledge. 



NORM OF SPACE DETERMINATIONS 171 

Facts have been collected and hypotheses proposed to 
synthesize them and yet these have had to give way for 
others, and others still, until the data of scientific knowl- 
edge to-day are quite different from what they were 
in earlier days. And yet permeating the scientific 
knowledge of all times has been the golden thread 
of reality, and of all facts and systems of facts which 
man has successively assumed and surrendered nothing 
has remained but the reality; indeed, nothing could 
so remain, but reality. So it is with air phenomena 
with which consciousness has to deal. This perhaps 
is due to the fact that the mind interprets phenomena 
in accordance with the quality of its awareness, and 
as consciousness is a variable quantity, its standards of 
interpretation will likewise vary. Each new scope of 
awareness, after this manner, yields higher and more 
exact standards of interpretation. And then, pro- 
gressing in awareness from the segment to the whole 
a fuller view of the phenomena as well as of reality 
itself is gained and also a more comprehensive judg- 
ment of the relations which exist between the segment 
and the whole. In other words, as the scope of 
consciousness widens it becomes more and more ap- 
parent that what was first thought to be a separate 
segment is in reality identified with the whole in an 
indissoluble manner. For the Thinker is then not 
only aware of the segment as such, but he is also 
conscious of the fact that it has definite relations with 
the entirety and that what he needs is merely a more 
extended consciousness. 

In denying the existence of the four-space or 
spaces of w-dimensionality as described and defined 
by geometricians, we do not thereby deny the existence 



172 THE MYSTERY OF SPACE 

of a plane of consciousness which is as much unlike 
the conditions of the tridimensional world as it is said 
to be unlike the four-dimensional world; but what we 
do deny is that such a higher plane of existence has 
necessarily to be conditioned by such characteristics 
as the metageometricians have proposed. It is main- 
tained that there is no basis in consciousness for a 
world of four dimensions; that the consciousness has 
no tendency for action in four-space. Neither has 
matter nor life any inclination or potency to behave 
in a four-dimensional manner. It is indeed more ra- 
tional to suppose that there is a higher plane, in fact, 
a series of higher planes, in which the thread of 
realism is continuous, not broken as it necessarily 
would have to be in extending to hyperspace, nor 
curved as in a manifold; that this series of subtler 
and finer planes of consciousness are merely an elonga- 
tion of our three dimensional scope of realism. It, 
therefore, remains only to master the phenomena of 
each in just the same manner as we have, in a measure, 
mastered the phenomena of tridimensionality. For it 
is easily conceivable that the quality of consciousness 
is such that it may adapt itself to a far wider range 
of possibilities than may be discovered in hyperspace 
and still be a tri-space quantity. 

It is believed, however, that in all the new and 
higher planes of consciousness tridimensionality is the 
norm both of the phenomena and of the reality 
peculiar to them. And that, being such, does not 
change or vary, but is a fixed quantity regardless of 
the plane of consciousness. Furthermore, it is believed 
that the highest state of consciousness in the entire 



NORM OF SPACE DETERMINATIONS 173 

kosmos could easily exist, and does so exist, upon the 
basis of three-space as the norm of its extent. 

A sharp line of demarkation should be drawn be- 
tween the reality which is life and consciousness and 
that which belongs to the realm of phantasy. For it 
is the prerogative of the intellect to create, out of 
the remains and deposits which it finds in the pathway 
of life, whatsoever it wills. This it does continu- 
ously; but it scarcely can be expected that such crea- 
tions shall be endowed with the same dynamic char- 
acter as that which life bestows upon its creations. 
The creations of the one are merely dead carcasses 
while those of the other are vital and real. Between 
them the same marked difference exists as between 
the growing tree and the lumber which the builder 
converts into a house. The organization which we 
witness when we look upon a building made of the 
dead body of a tree is not the same kind of organiza- 
tion as that which we see when we view the living, 
growing, vital tree. The dead tree is a deposit of 
life cast off by it when it passed on. Whatever the 
intellect can do in disposing of the remains of the 
tree-life is conventional and artificial. If it convert 
it into an edifice it will then bestow upon it a sort of 
consistency which is quite sufficient for all purposes. 
But the consistency which holds the organization of 
an edifice together is not the kind of consistency which 
holds a living tree together. In fact, there is a con- 
sistency that is not consistent. Such is the consistency 
of metageometry. It is self-consistent and yet incon- 
sistent with the consistency of the kosmos and its norm 
of being which is consciousness. 

Self-consistency is one thing and kosmic con- 



174 THE MYSTERY OF SPACE 

sistency is quite another. It does not necessarily fol- 
low that because a given scheme of thought is con- 
sistent in all its parts that it is also consistent with 
universal truth or with life. This very vital fact was 
overlooked by Gauss and all those who followed in 
his wake when he discovered that his Astral Geom- 
etry was consistent throughout in all its parts. There 
is only one norm of truth and that is kosmic con- 
sistency. It matters little that a thing shall be self- 
consistent; it matters much whether it is consistent 
with the universal standard. It has been shown to 
be logically possible to elaborate at least two different 
systems of geometry, namely, the geometry of the 
acute angle and that of the obtuse, which, while each 
of them is self-consistent throughout, are nevertheless 
inconsistent with each other and with the geometry 
of the right angle (Euclidean). This, it would seem, 
appears to be sufficient for the invalidation of either 
one or both of the non-Euclidean systems of geo- 
metric thought. Indeed, if it can be shown that the 
Euclidean geometry is more representative of the true 
approach to the norm of space-genesis and of creation 
so far as its mode of manifestation is concerned, and 
consequently true of the norm set up by consciousness, 
the rejection of both systems of non-Euclidean 
geometry seems to be thoroughly warranted. But this 
is obvious and requires no demonstration nor com- 
ment to make it clear. We have only to ask ourselves 
whether it has ever occurred to us that consciousness 
has either a tendency to or adaptability for action 
in a curvilinear manner; or, if when we contemplate 
ideas or idea-relations we have the impression of per- 
ceiving a curvilinear or manifold tendence in them 



NORM OF SPACE DETERMINATIONS 175 

either of a positive or negative nature, and also 
whether it has been observed that our thought proc- 
esses naturally assume four-dimensional attitudes. If 
we find that such a query must be answered negatively, 
and indeed we must so find, then, we have no basis 
for the assumption that any one of the systems of non- 
Euclidean geometry is valid either for the present 
status of consciousness or for a future existence, since 
it is true that the future is but an elongation of the 
present. Evolution is to bring no radical changes in 
the norms of reality; it has merely to deepen and 
widen and make more intense, efficient and compre- 
hensive the present scope of our consciousness and 
thereby, while the Thinker is passing from one degree 
of realism to another, to bring him into a clearer con- 
ception of what his own limited scope of motility 
means to the whole. 

The four-space is a mathetic divertisement. That 
is, it cannot be said to lie in the direction of a straight 
line which proceeds either in a forward or lateral 
direction. Neither does it lie in a plane which is 
vertical or horizontal to the sensorium. It is, there- 
fore, a fractural departure from any conceivable tri- 
dimensional direction, a geometric anomaly. Evolu- 
tion, despite the minor aspects of its movement, 
undoubtedly proceeds in a straight line and not by a 
zigzag nor discontinuous line and hence it is irrational 
to assume that it will, when it passes on to the next ad- 
vanced stage, emerge into the realm of the four-space. 
For the so-called hyperspace of geometry cannot by 
any standards of reference be said to lie in the plane 
of any straight line which can be described in three- 
space. If life is to evolve more efficient forms and 



176 THE MYSTERY OF SPACE 

if the forms are to evolve into more perfect organiza- 
tions and mind and consciousness to become more 
intense and comprehensive expressions of the divine 
mind of the kosmos it is certainly not in the domain 
of hyperspace that these shall find the substructure of 
their higher development; but, if at all, it shall be 
found, as in all times past, in the realm of perceptual 
space where bodies are said to have three and only 
three possibilities of motion. 

What then is the significance of the more than a 
thousand years of mathematical labors; of all that 
has been said and done in an endeavor to bring into 
the popular consciousness a conception of hyperspace? 
Is it a question of "Love's 'Labour's Lost?" Or is it 
a mere prostitution of mathematical talent? To an- 
swer these queries is the burden of this treatise and 
it is hoped that as the text continues the reader may 
be able to arrive at his own conclusions as to the 
relative value of the work of the mathematicians in 
this respect and be able to judge for himself the true 
significance of it all. 

The specific value of consciousness as a deter- 
minative factor in space-measurement has been recog- 
nized by all who have sought to arrive at a logical 
justification for the conception of four-dimensionality 
by analogous reasoning. The existence of the unodim 
with consciousness limited to a line or point has been 
assumed and it has been shown how greatly such a 
being would be handicapped by his limited area of 
consciousness, it having been proposed to confine his 
consciousness to one dimension. An unodim would, of 
course, be entirely unaware of any other dimension 
than that in which he could consciously function. So 



NORM OF SPACE DETERMINATIONS 177 

tHat with respect to his own consciousness no other 
dimension would be necessary for the continuance of 
his life processes. He might live his life without any 
knowledge even of any limitations or barriers to other 
things higher than those of his plane. He would be 
content to exist in the one-space and enjoy the benefits 
which it offered. He could have no notion of the 
two-space, but it has been allowed that a super-unodim, 
an unodim metageometrician, if you please, could 
reason out a mental conception of what the two-space 
might be. Passing on to a space of two dimensions, 
the domain of the duodim, a greater freedom of move- 
ment is allowed and instead of being able to function 
in only one dimension the inhabitants of this plane 
would find themselves able to move about in at least 
two directions. Consciousness would accordingly enjoy 
a more comprehensive scope. But in a manner similar 
to that used by the unodim metageometrician it is held 
that the duodim could get a conception of the three- 
space by analogous reasoning and so gradually become 
conscious of a higher degree of spatiality than his 
own. In the conscious reasoning of both, however, 
is the condition of perpendicularity. That is, it must 
be assumed by both the unodim and the duodim that 
the new dimension must lie in a plane perpendicular 
to their space. So, the unodim would postulate that 
the two-space must lie in a direction at right angles 
to his space, and yet he would not be able to indicate 
the direction owing to his ignorance of any experience 
that would acquaint him with the new space as well 
as the want of possibility of motion therein. Simi- 
larly, the duodim would arrive at a conception of 
three-space. Thus, it has been argued that tridims, 



178 THE MYSTERY OF SPACE 

or people living in our tridimensional world, could, 
by using a like line of argument or reasoning, arrive 
at a conception or understanding of the four-space, 
which, of course, must also lie in a direction at right 
angles to three-space. 

The implications of this mode of thought show 
how thoroughly the mathematician recognizes the 
limitations which consciousness imposes upon our 
knowledge of the world and the subtler conditions 
about us. But, moreover, it is even obvious to all who 
stop to think about it; for it can readily be seen how 
little those things which do not enter our scope of 
awareness affect us .either physically, mentally or spir- 
itually. But no one can be so bold as to deny utterly 
that anything exists but what is found in our con- 
sciousnesses. It is even true that in the great centers 
of population where people are compelled to live many 
families in the same house, it is the usual thing for 
these individual families to live in complete forgetful- 
ness of all the others in the house and live their lives 
so completely that it would be exceedingly difficult to 
measure the effect the one has upon the other. The 
mathematician, as is shown by the hyperspace move- 
ment, recognizes that there are planes of supercon- 
sciousness the nature and character of which persons 
confined to limited areas of consciousness can have no 
knowledge and may only arrive at that knowledge by 
serious thought and contemplation. In other words, 
they tacitly admit the existence of higher planes of 
consciousness as well as the necessity of elevating the 
personal consciousness in order to comprehend them. 
Although it was not expressly allowable in the analogy 
of the unodim ) it is nevertheless one of the strongest 



NORM OF SPACE DETERMINATIONS 179 

implications of the process of reasoning that the 
unodim could have easily raised the plane of his con- 
sciousness by continuing his researches until he, too, 
became conscious of the three-space, mathematically, 
as well as the two-space. For it was not necessary 
for him to raise the plane of consciousness in order 
to contact the two-space. He had need only to widen 
it. But in order to comprehend the mathematical 
three-space it would be necessary for him to elevate 
his consciousness. 

The fundamental error in the foregoing line of 
thought rests in the fact that awareness in the human 
family has not developed in the manner outlined. 
The human species has not come into conscious rela- 
tions with the three-space by outgrowing the one-space 
and the two-space in succession. The fact of the 
matter is that when consciousness first dawned it must 
have encompassed all three dimensions simultaneously 
and equally and there is nothing to indicate that its 
rise was otherwise. Then, specifically there is no 
evidence that the evolution of consciousness has pro- 
ceeded in a rectangular manner. Indeed, there is un- 
doubtedly no warrant for the assumption that it has 
progressed in ways that are mathematically determin- 
able at all. The question very naturally rises in view 
of the above as to the relative value of mathematical 
knowledge in the scheme of psychogenesis. Can 
mathematical knowledge or laws be said actually and 
finally to settle once for all time any question in which 
consciousness or life enters as a factor? Upon the 
response to this question hinges unanswerably the de- 
cision as to the category which mathematical knowl- 
edge should by right occupy in the entire schematism 



180 THE MYSTERY OF SPACE 

of life. If it can be successfully maintained that final 
judicative power abides in mathematics in the deter- 
mination of these questions, then it would be useless 
to struggle against the fiat of mathematics and mathe- 
maticians; verily, we should be compelled to accept 
nolens volens all that mathematicians have devised 
about hyperspace and its connotations. If, on the 
other hand, it can be shown that no such judicative 
power inheres in mathematical knowledge we shall 
then be able to establish for mathematics a true cate- 
gory and to dispose of the hyperspace movement in 
a manner that shall at once be logical and necessary. 
That the discovery of hyperspace by the mathe- 
matician is merely an aspect of a general forward 
movement in the evolution of consciousness can be 
shown by a brief correlative study. Hyperspace is the 
artificial symbol of a higher and more extensive realm 
of awareness. For it cannot be doubted that to be 
able to think in the terms of hyperspace, to study the 
various relations and interrelations upspringing from 
the original premises, actually to become conscious in 
the hyperspatial realm thus constructed, requires a 
different species or quality of consciousness than that 
required for ordinary thinking. The period covering 
the rise of artificial spatiality is contemporaneous with 
the rise of the phenomena identified with the spiritual 
life of Swedenborg; for during the same time he 
began a series of visions which revealed to him great 
knowledge of the unseen and supersensuous realities 
of life and existence. His consciousness was being 
flooded with the light from so-called celestial spheres 
and he was gradually becoming conscious of a "new 
dimension," a new space, a higher world that is alto- 



NORM OF SPACE DETERMINATIONS 181 

gether unlike the world of the senses. During this 
period, too, Dante, the great kosmic seer began to 
jot down the results of his "hyperspace" experiences, 
after which he wrote his Divina Commedia in which 
he describes more or less minutely some of the char- 
acteristics of the hyperspace domain which was re- 
vealed to his consciousness as he saw and interpreted 
it. Both Swedenborg and Dante being deeply re- 
ligious and pious-minded had their reports of the new 
world colored by their own mental experiences and 
proclivities. Plato had at an earlier day set down 
what he conceived to be the ethical and civic char- 
acteristics of the new age, the Utopia of mankind 
living on a higher plane. It was during these days 
that the doctrine of evolution was born, although it 
remained for Darwin to formulate and buttress it 
with a stupendous congeries of facts. Martin 
Luther, the great religious reformer, likewise con- 
tacted the radiating light-glow of a higher conscious- 
ness into which the race was coming but of which 
only the foremost were able to get advance glimpses. 
Kant, one of the peerless leaders of the vanguard 
of humanity, at this time also, conceived and wrote 
his Critique of Pure Reason which is likewise an evi- 
dence of the upliftment of his consciousness on the 
side of pure intellectuality and the commencement of 
a general period of illumination. And then, later, 
but embraced within the same period, artists began 
to get glimpses of this higher consciousness which 
showed itself in a new and strange departure in art. 
In rapid succession new schools sprang up and came 
to be known as the "cubist," "post-impressionist," 
"futurist," "orphimist," the "synchromist" and the 



182 THE MYSTERY OF SPACE 

"vorticist." Art really began the search for the 
"plastic essence of the world" trying to portray its 
conception of the "real image of the spirit" of the 
world. Color acquired a new kind of splendor and 
painting gave birth to a new intrinsic beauty of mate- 
rial and sheer loveliness of texture. All of which 
were evidences of an intellectual up-reaching in re- 
sponse to the sharp appulsions from above. Darwin's 
mind, being of scientific bent, saw and interpreted 
everything in the terms of materialistic science; but 
there is no doubt but that the expansion of the area 
of awareness which his mind experienced in his great 
conception of evolution as a continuous process and 
all that it implies thereby was a result of the universal 
appulsion of the human intellect against the new 
domain of consciousness. And Kant's conception of 
space in general perhaps may be said to have been the 
seed-thought for the metageometrician. 

But thus it will be noted that in all the cases men- 
tioned in the foregoing there is always present the 
personal element of the investigator, and that the re- 
ports of each of these have been colored and char- 
acterized by their individual consciousness and expe- 
riences. That all reports would agree with respect 
to details connected with the new domain of conscious- 
ness could scarcely be expected owing to the wonder 
and bewilderment which seize the intellect under such 
circumstances. No implication that the mathemati- 
cians have been unduly excited by what they have dis- 
covered after years of patient research in this direc- 
tion is indicated by the foregoing observations; but 
it cannot be denied that the enthusiasm of the moment 
and the consequent minimization of all other phenom- 



NORM OF SPACE DETERMINATIONS 183 

ena but the special line being investigated serve very 
effectively to obscure the mental vision of the more 
partisan. It perhaps is sufficient that the investigator 
should set down in as orderly manner as possible the 
things which he conceives, and that he should interpret 
them according to the standards of his own intellect. 
More than this cannot be expected. Moreover, it 
usually suffices that the future investigator, far re- 
moved from the beclouding influences of partisanship, 
who successfully raises his consciousness to that 
higher plane shall be able to synthesize the findings 
of all and thereby with the aid which comes to him 
from a more advantageous position arrive at sounder 
views and a more reliable judgment. 

It will thus be seen that the metageometrician's 
method of interpretation is no more entitled to final 
credence and general acceptance than that of the 
spiritualist, Swedenborg, or the occult seer, Dante. 
For in their best moods and at their highest points 
of mental efficiency these have only succeeded in 
vaguely symbolizing what they have conceived of the 
realities of the supersensuous realm in terms of their 
own experiences. Is there any more cogent reason, 
then, for accepting the analyst's conception of a world 
of hyperspace peopled with ensembles, propositions, 
spaces of ^-dimensionality and other mathetic con- 
trivances than the Inferno of Dante, inhabited by 
hideous shapes and repellent denizens, the remains of 
ill-spent earth-lives or Swedenborg's Celestial Realm, 
wherein dwell numerous beings of celestial character 
performing various tasks in the work of the world? 
These observations should not lead the reader to come 
to the conclusion that the visions of I)ante and 



i8 4 THE MYSTERY OF SPACE 

Swedenborg are deemed to be more worthy of 
credence than mathematical knowledge when that 
knowledge is limited to the sphere where it rightfully 
belongs; but the proper view is that which would make 
it appear that it is the way these widely differing 
workers interpret what they have seen; that it is the 
adaptation of the unseen realms to the peculiarities 
of the mentalities which observe them. The mathe- 
maticians have simply portrayed as well as they could 
their conception of the new stage of consciousness and 
its contents, and following the modus vivendi of all 
intellects have interpreted these things in the terms of 
mathematics, merely because mathematics constituted 
the best available symbology at hand for the purpose. 
Similarly, the painter sees a new world of color; the 
politician, a new era of political freedom; the religious 
enthusiast, a new religious conception; the scientist, a 
new condition of matter and energy, and so on, to 
the most ordinary mind, each sees something new 
while at the same time is necessarily limited to the 
confines of his own mentality when he comes to in- 
terpret what he sees and conceives. Hence, there 
would appear to be only one way to regard all these 
advances and that is by synthesizing them, by corre- 
lating, and by tracing them to a common source, and 
finally by seeing them as one general forward move- 
ment of intellectual evolution. 

Man, the Thinker, who in essence is a pure in- 
telligence, has two mental mechanisms or organs of 
consciousness. One of these is the brain-consciousness 
or the egoic. It is so called because the brain is its 
organ of expression and impression. It manifests 
through the brain and uses it to further the various 



NORM OF SPACE DETERMINATIONS 185 

objective cognitive processes. The brain-consciousness 
is a child of the physical body and its life is intimately 
identified with the life of the body. This conscious- 
ness may be called the a posterioristic mechanism or 
organ of the Thinker and is therefore his means of 
interpreting the phenomena of the objective world. 
Cell-consciousness is a phase of the ergonic functions 
of the a posterioristic mechanism. The other organ 
of consciousness is an aspect of the intelligence of the 
Thinker himself and perhaps may be said to be the 
active, organized portion of that intelligence. It is 
separate and distinct from the a posterioristic con- 
sciousness yet sustaining a substructural relationship 
with it, being the source of the egoic or brain-con- 
sciousness. It may be called the a priori conscious- 
ness. Its roots are buried deep within the heart of 
the space-mind and it is therefrom nourished and de- 
veloped by what it receives in the way of intelligence. 
It is the intuitional faculty; knows without being 
taught; conceives without reason; interprets according 
to the norms of the space-mind or the divine mind of 
the kosmos. It always resides on a higher plane than 
that of the brain-mind or consciousness, only at rare 
intervals being able to contact it with flashes of its 
own intelligence as intuitions. 

The a priori consciousness being the intuitive 
faculty of the Thinker is, therefore, a phase of his 
mental life on a higher plane than the sensuous. All 
its conceptions constitute the a priori knowledge of 
the brain mind so-called. The a priori faculty of man's 
higher consciousness gives the character possessed by 
that form of knowledge known to philosophy as the 
a priori. So that the a priori has a more substantial 



1 86 THE MYSTERY OF SPACE 

basis than has hitherto been surmised. It is not only 
that which may be said to transcend experience but 
that which is the organ of contact with the supersensu- 
ous realities as well as of expression through the 
brain-consciousness. 

The mind's method of apprehending objective phe- 
nomena is not a direct process but an indirect process by 
virtue of which neurograms or nerve-impacts registered 
in the brain are interpreted. External sense-impressions 
are, of course, conveyed to the cortical area by means 
of appropriate vibrations which traverse the lines of 
the neural mechanism. These are recorded in the 
brain areas just as a telegraphic communication is 
registered in the apparatus of the receiving end, and 
in being so, they make terminal registrations which 
man, the Thinker, interprets after a psychic code which 
has been built up empirically. That is, he comes to 
know that certain rates of vibration and certain 
peculiarities therein mean certain things when re- 
ferred to the sensorium. He then interprets accord- 
ing to this experience the symbolism of all neuro- 
graphical impressions. But it is obvious that under 
such circumstances, where the interpreter is far re- 
moved from the thing itself and finds it necessary to 
interpret rates of vibration or symbols in order to 
arrive at a knowledge of the intelligence which is con- 
veyed to him, the chances of inadequate conception 
are very great indeed. In fact, it is not possible 
through the use of neurographical symbols alone to 
get any complete notion of the phenomena considered. 
And thus there stands between the Thinker and abso- 
lute knowledge a barrier which prevents his arriving 
at a state of certitude in his knowledge of the world 



NORM OF SPACE DETERMINATIONS 187 

of sensible objects. It is, however, a barrier which 
will always remain, checking ever his approach to 
finality in his understanding of the universum of ap- 
pearances. 

A markedly different condition obtains in the realm 
of the a priori or intuitional for the reason that the 
barriers which inhere in the neurographical or a 
posterioristic method are absent and the Thinker has 
a more direct approach to the objects of cognition. 
Hence the chance for error is very small indeed. This 
will account, therefore, for the superiority of the in- 
tuitional over the rational or the perceptual. Indeed, 
it is doubtful whether the purely rational possesses 
any value whatsover when its modus vivendi is un- 
sanctioned by the intuitional. 

Else why can we not be certain that the results of 
our rational processes are correct at all times? Is it 
not because we lack the power to perceive whether 
our premises are correct in the first place? Quite 
truly. For if the Thinker can intuit the necessity and 
certitude of any given premise it follows that the 
consequences of that premise are true. It would, 
therefore, appear that the more the intuitional faculty 
is developed the clearer will be our perceptions not 
alone of abstract values but of objective things them- 
selves. Further, it is doubtlessly true that the more 
the space-mind is developed in the human race the 
deeper will become our perceptions of the essential 
be-ness of things so that whatever may be the presenta- 
tions of the space-mind to the brain-mind they will be 
by far more accurate than the impressions we receive 
through the latter as a medium of apprehension. It 
is but natural, however, that in the present more or 



188 THE MYSTERY OF SPACE 

less chaotic condition in which the faculty of the 
intuitional is found it should be difficult even to inter- 
pret its presentations accurately. It is perhaps due 
to the fact that we are unused to its deliveries and 
mode of registration as well as to the fact that it has 
been overshadowed by the intellectual or rational 
faculty. But the mere fact that it is present and func- 
tioning, even if but rudimentarily, is evidence of its 
potentiality and the possibility of its future develop- 
ment to a still higher degree of efficiency. 

There is no doubt but that the original impulse 
which resulted in the creation of the faculty of per- 
ceptibility in the Thinker also marked out the metes 
and bounds of our entire range of perceivability which 
includes not only the intuitional but something higher 
still. There is no doubting either the obvious 
fact that these metes and bounds cannot have been 
other than rudimentary or general lines of denotation, 
and that the work of their further elaboration and 
refinement is a matter of evolutionary detail. For if 
we assume that the general principles of evolution are 
true we immediately recognize the cogency of this 
view. That which we now call the hand has not 
always been the perfect instrument that it is nor has 
the ear always been so keenly adjusted as at present. 
It has required undoubtedly many million of years 
for the eye to reach its present degree of complexity 
and adaptability. Yet in all these cases the different 
organs existed in potentiality from the beginning; the 
metes and bounds of the hand, the ear and the eye 
were laid out primordially. Evolution has specialized 
and adjusted them to environments and needs. Thus 
it will be seen that while the intuitional faculty was 



NORM OF SPACE DETERMINATIONS 189 

designed for manifestation from the beginning it has 
nevertheless required ages for its appearance even in 
the most rudimentary fashion. 

Almost the entire content of human knowledge is 
based upon assumptions or hypotheses; in fact, is but 
a mass of these, and especially is this true of mathe- 
matics, science and philosophy. Of course, there are 
certain minor observable facts which by reason of the 
seeming permanence of their existence have been 
eliminated from the category of assumptions. But 
even this elimination when it is traced to its depths 
may be found to be erroneous, and perhaps after all, 
when we have really begun to know something of the 
reality of things, may have again to be placed in this 
category. And then, too, the hypothetical nature of 
our knowledge is due largely to the Thinker's method 
of contacting the objective world which is the subject 
of his knowledge. It is because it is necessary for him 
to interpret the neurographical symbols which sense- 
impressions make in the brain matter according to a 
psychic code that renders his knowledge of things in 
general hypothetical. His interpretations are based 
upon an assumed value which experience has taught 
him to give to each neurogram. But even when his 
interpretations leave nothing to be desired in respect 
to their accuracy of apprehension of what the neuro- 
graphical vibration implies there is that further bar- 
rier to his cognition of reality which is due to his re- 
mote removal from the object itself and the consequent 
extreme difficulty, if not present impossibility, of 
identifying his consciousness with the essence of the 
objects which he contemplates. 

When the Thinker's consciousness is presented 



i 9 o THE MYSTERY OF SPACE 

with a neurograph of say, a cube, it is not the cube 
itself which he contemplates or observes; it is the 
neurograph or psychic symbol which the sense-impres- 
sions make in the brain. His consciousness deals not 
with the object but with the symbols. It is true that 
when he verifies one neurograph by another, as the 
scopographic or sight impressions by the tactographic 
or touch impressions it is found that the delivery thus 
determined is a true enough representation. It is also 
true that the Thinker, as a rule, does not accept a 
neurograph as valid until it has been verified by at 
least one or more presentations through his outer 
sense organs. It occurs, therefore, that all such de- 
liveries are verified and corrected by one or more 
sense witnesses before final acceptance by the Thinker; 
but even then it cannot be said that his notions thus 
gained are in all respects correct and true to the 
standards set up by the brain-consciousness not to 
mention higher forms of consciousness. And then, 
when we consider that in addition to the numerous 
chances for error which naturally inhere in this method 
of cognition it must also be apparent that the Thinker's 
approach to the reality of things is much impeded by 
his separation therefrom, the unreliability of our 
ordinary methods of cognition is much emphasized. 

But aside from the egoic or brain-consciousness 
there is the higher consciousness of the Thinker him- 
self. For the brain-consciousness is merely his method 
of regarding and comprehending the neurographical 
deliveries, the psychic code by which he systematizes 
and organizes his cognitions or impressions of the 
sensible world. This higher consciousness constitutes 
the faculty a priori for the Thinker on a higher plane 



NORM OF SPACE DETERMINATIONS 191 

of existence, and because it deals with elements alto- 
gether unlike those which make up the data of brain- 
consciousness is, accordingly, less liable to error in its 
judgments of the supersensuous presentations than is 
the objective or brain-mind. In fact, it is difficult to 
conceive of a state or conditions wherein, supported 
as this view contemplates, the intuition should err in 
judgment. Viewed from the standpoint of external 
impedimenta, this condition may be said to be due to 
the absence of sensuous barriers which would other- 
wise prevent the near approach of the Thinker's con- 
sciousness to the essence of things which is the object 
of his consciousness on this higher plane. Directly, 
however, it is undoubtedly due to the fact that, follow- 
ing the lead of life itself, yea, as the veritable hand- 
maid of life, it cannot err where life is concerned. 
When dealing with notions a priori or intuitograms 
the Thinker is relieved of the onerous necessities and 
limitations incident to the examination and determina- 
tion of neurographic symbols registered in the brain 
cortex and so is free to study, to examine and judge 
at first hand the impressions which are received from 
his own plane of intuition. The difference is about the 
same as that which should exist between the methods 
of communication between two telegraphic operators 
when in one instance they would have to depend upon 
the deliveries conveyed over the wires, while in the 
other, when they stood face to face with each other, 
they could communicate by direct conversation. In the 
one case the method of communication is direct and 
simple, while in the other it is indirect, circuitous and 
complex. It can, therefore, be readily seen that in 
all cases where the approach is made in a direct, simple 



i 9 2 THE MYSTERY OF SPACE 

manner the probability of error is much less than in 
cases where the intellectual approach is less direct and 
more complicated. Hence in drawing conclusions as 
to the relative importance of the two mechanisms of 
consciousness, the a posterioristic and the a priori, it 
is necessary to bear in mind the comparative superiority 
of the one over the other as a means of cognition. 
It matters little that the intuitional faculty is not so 
well developed as the tuitional because it is but natural 
that inasmuch as the Thinker's needs are subserved 
in the sensuous realm by the tuitional consciousness it 
should, from more active use, gain somewhat over 
the intuitional in facility of expression and general 
utility. And the more so, because the two faculties 
serve different purposes; one is attuned to receive im- 
pressions from a subtler plane while the other is fitted 
for contact with the phenomenal universe; one is re- 
lated to the conceptual while the other is related to 
the perceptual. They differ not only in function, but 
in nature as well. There is, of course, a natural bar- 
rier consisting of the inherent limitations of each 
faculty which prevents the full mergence or unification 
of the two states of consciousness so that there exists 
a state of consciousness the data of which the brain- 
mind is unaware, it being able only at rare intervals 
even to receive so much as slight impressions from it 
in the nature of intuitional flashes or inspirations and 
the like. Viewed in this light it would appear that 
the cognitions which are most truthworthy are those 
which are presented by the intuitional faculty because 
they are nearer to the essential reality of things ; they 
have to do more specifically with the nature of that 
which appears while the tuitional mind can only regard 



NORM OF SPACE DETERMINATIONS 193 

that which is the appearance. Herein lies the whole 
difference. 

The natural outcome of this division of labor be- 
tween the tuitional and the intuitional is the establish- 
ment of the fact of man's relationship both to the 
phenomenal and the real; that in his psychic nature 
must reside the faculty of apprehending the real and 
that he shall one day awaken into activity this now 
latent faculty whereby he may make a direct and naked 
contact with reality. 

If it be true that, as Plato said, God does 
geometrize, and that the divine geometry, as will ap- 
pear, is based upon a system, an alphabet which taken 
together are the point . , the line , the triangle 

/\ , the square and the circle ( \ then, 

we have in this geometric alphabet the very secret of 
the divine geometry. With these, and in the kosmic 
laboratory of chaogeny, the Creative Logos has 
measured off the limits and confines of space; with 
them He has traced out its dimensions the archeo- 
logical evidences of which we may view in the space- 
mind itself; and with them he has established the 
manner of its appearance to the Thinker. In dimen- 
sions, three, and yet not three, but one, Space, the 
eternal progenitor of all forms and energies, having 
received the divine fiat in the beginning that thus far 
it should extend and no further, persists in faithful 
obedience to the law of its being — tridimensionality. 
It must be so because it is thus sanctioned by the high- 
est faculty in man that can render judgment thereon. 
If tridimensionality inhere in the space-mind, as the 
law of its being and in the intuitional consciousness 



i 9 4 THE MYSTERY OF SPACE 

as the norm of its essential nature and as the easiest 
and simplest expression of the tuitional mind, how can 
it be gainsaid that these considerations obviate the 
necessity of the mathetical hyperspace? 

If the reality of things is hidden from us and if 
we are, therefore, unable to perceive their real essences 
it is because our mode of thought and our conscious- 
ness have obscured our vision and limited us to this 
state of paucity of perception. It is not because reality 
is itself a hidden, inscrutable quantity nor that its 
modus vivendi is "unknowable"; but because we be- 
ing multiformly limited, "cabined, cribbed and con- 
fined" are resultantly lacking in the power to discern 
that which otherwise would be most obvious to us. It 
may well be set down as axiomatic that when, in the 
process of our thinking, we arrive at the inscrutable, 
the unknowable and the infinite, it is evident that our 
thought processes are dealing with a form of realism 
which is higher and beyond the possibilities of our 
loftiest thought-reaches. And in order to symbolize 
to itself this condition the intellect poses such terms as 
"inscrutable," "unknowable" and the "infinite" simply 
because that is the best it can do. Hence when it is 
said that space is infinite it is apparent that the mind 
recognizes that when it contemplates space it is deal- 
ing with something whose degree of realism transcends 
its powers of comprehension. Infinity is a relative 
term, and in fact, decreases in extensity in the propor- 
tion that the consciousness expands and comprehends. 
It is not unlikely that should the intellect one day 
discover that it had awakened into union with the 
space-mind it would immediately reject its preconceived 
notion of the infinity of space. But we need not wait 



NORM OF SPACE DETERMINATIONS 195 

until the coming of this far off event in the path of 
psychogenesis ; for we can here and now perceive with 
what must be a higher faculty than the intellect the 
verity of this conclusion. 

But certain it is that the intellect, in the pride and 
arrogance of its traditional heritage, will not without 
a great struggle yield the ground and prestige it has 
held for an aeon of time; and in vain does the intui- 
tion serve notice of dispossessal in these premises; but 
however stubbornly fought the battle, however tena- 
ciously held the position time will discover the weak- 
ening of the intellect's hand. Death for the intellect 
may ensue as a result of the conflict but it will be a death 
wherefrom it will arise, quickened, revivified and up- 
lifted by its disposer, the intuition, upon the remains 
of its dead self to a higher and grander state than it 
has ever enjoyed before. 

Space is not static. It is dynamic, potential and 
kinetic. It is a process, a becoming. Its duration as 
a process is never ending. Its extensity is limited and 
finite. The so-called infinity of space is one of the capi- 
tal illusions of the intellect which can only be removed 
by an expansion of the consciousness, by a mergence 
of the individual consciousness with the space-con- 
sciousness. In the ever-widening circle of the indi- 
vidual consciousness lesser realities give way to greater 
as the darkness recedes from the light — the lesser 
appearing in comparison with the greater, as the con- 
sciousness broadens, as matter to spirit, as night to 
day or as limitation to non-limitation. Thus the most 
solid facts and conditions of our limited life are but 
the shadows of the deeper realities which shall be 



196 THE MYSTERY OF SPACE 

revealed to the Thinker in the days of his larger and 
more glorious life of freedom from limitations. 

And now it will appear that the whole fabric of 
our knowledge shall have to be reduced to the bare 
warp and woof; for nothing is real but these. It 
is as if the Thinker, using the tuitional mind, had 
been in all times past studying the design woven 
in the surface of a very thick plush carpet. There 
are the warp and woof, the long vertical threads 
which make the plush and the intricate design 
appearing on the surface. Our knowledge may be 
likened to the design. It represents the contents of 
our knowledge. We have not even so much as begun 
the study of the nature of the vertical threads as they 
appear beneath the surface to say nothing of begin- 
ning the study of the warp and woof. The warp and 
the woof are the realism of the kosmos; the vertical 
threads are the roots and stem of the phenomenal 
world; the design is our sensible world as it appears 
to the intellect. The life of the intellect has been 
spent in contemplating this design; while of the hands 
which wove the carpet, of the mind which directed 
the hands and of the spirit which vitalized all, it knows 
nothing nor indeed can it know anything. Where shall 
we say are those hands, that mind and that spirit 
which made the carpet possible and an actuality? In 
vain do we search among the remains, among the soft, 
glistening threads of the carpet or among the in- 
tricacies of the design. For they are not there. They 
have passed on. The intellect looks at the design or 
at the vertical threads and because it is unable to fol- 
low them to their source, it decides that they are in- 
finite, inscrutable and unknowable. But not so. All 



NORM OF SPACE DETERMINATIONS 197 

that is required are eyes to see and a mind (or shall 
we say a mind vitalized by the intuition) trained to 
discern the threads as they point upward with their 
termini firmly rooted in the warp and woof of the 
fabric. But we must first master the design, and then 
turning to the threads, master them. Then shall the 
doors of kosmic reality swing wide and the Thinker 
shall be ushered into the eternal palace of kosmic 
realism wherein he shall find the great secret, the 
heart, the purpose, the beginning and the end, the very 
nature of things-in-themselves. 

The nature of every degree or condition of real- 
ism is so constituted that its qualities, characteristics 
and limitations are exactly adequate for the satisfac- 
tion and fulfillment of all the requirements and needs 
of every possible state of normal consciousness. So 
that each degree of reality and each state of normal 
consciousness is sufficient and complete in itself and 
mutually satisfies the necessities of each other. The 
substratum of reality or life which extends from the 
heart of the kosmos to the extreme limits of the phe- 
nomenal universe exists in degrees, not discrete, but 
continuous. And these merge into one another by in- 
sensible stages. Such is the imperceptible continuity 
of the whole as each degree is gradually immerged 
into the other that only the limitations of conscious- 
ness itself make it to appear as if it were discontinuous. 
For every stage of realism there is a state of con- 
sciousness which answers to it completely and suffi- 
ciently. So both the state of consciousness and that 
of reality, manifesting at any given stage, seem to 
be complete and final for that stage. Realism or life 
and consciousness possess only a relative finality 



198 THE MYSTERY OF SPACE 

fashioned upon the necessities and requirements for 
any given state of being. Consciousness alone fixes 
the apparent limits of life; it also determines the state 
of our knowledge of life. And thus when the Thinker 
is confined to any stage of reality and congruent degree 
of consciousness it appears that what he there finds 
is ample for all his purposes. Accordingly he is con- 
vinced that that stage is the final consideration of 
his scope of motility. It is only when he is able to 
raise his consciousness to a point where he can contact 
higher realities that he becomes aware that there are 
higher stages in which his consciousness may manifest. 
This peculiarity of the Thinker's consciousness is ac- 
centuated when he allows himself to become wholly 
engrossed with a study of the phenomena of that stage 
in which he can consciously function. Hence it con- 
stantly occurs that men exhausting the study of the 
phenomenal find themselves floundering upon the beach 
of the outlying shores of consciousness where in sheer 
desperation they fall into the illusion that they have 
indeed reached the limits of manifested life and that 
beyond those limits there is no organized being. Un- 
conscious are they that in ever widening circles the 
fertile lands extend and await the awakening of their 
consciousness when they may till the fallow ground 
of this new domain and begin again the search for the 
ultimately real. 

With respect to the present powers of conscious- 
ness, it cannot be successfully controverted that the 
concept of tridimensionality of space is sufficient for 
all purposes. It must be so for it is not only an aspect 
of the phenomena of space but of reality as well. This 
fact is attested by the nature of mind that answers 



NORM OF SPACE DETERMINATIONS 199 

to the nature of space. Tridimensionality character- 
izes the entire extent of consciousness and life, and 
therefore, of space itself. This characterization may be 
traced to the very doors of the heart of space where the 
three become one. Nor would this conception be in 
the least vitiated if it were allowed that the mass of the 
phenomena of the supersensuous world, lying in close 
proximity to the sensuous world, does present itself 
to the consciousness in a four-dimensional manner and 
that the phenomena of a still higher plane present them- 
selves in a five or ^-dimensional manner to that state of 
consciousness which may be congruent with them; be- 
cause then we should be making allowances for the 
changes in phenomena and their mode of presentation 
to the consciousness which by no means implies a cor- 
responding change in reality or life. All phenomena 
are fashioned by the intellect. The phenomenal world 
is just what the intellect interprets it to be. It is that 
and nothing more. Its qualities, attributes and char- 
acteristics are such as the consciousness gives to it. It 
exists only for the purposes of the evolving conscious- 
ness. And, as an instrument of consciousness, its ex- 
istence is strictly subject to the evolutionary needs 
thereof. In that moment that the immediate needs 
of the consciousness shall no longer be able to find 
satisfaction in the phenomena of any plane of nature, 
in that moment the phenomena of that plane disap- 
pear, recede and are swallowed up in the maelstrom 
of eternal reality. 

In the gradual expansion of consciousness as it 
passes through the infinite series of grades of aware- 
ness meantime becoming deeper, broader and more 
comprehensive as it proceeds, there may be observed 



200 THE MYSTERY OF SPACE 

running through all these planes and orders that which 
is neither the phenomena of the various planes nor 
the consciousness; but which must be the substructural 
basis of both, remaining the same, unchanged and un- 
changeable. That is the thread of reality, the passage 
of life itself which is the eternal basis of all. Now 
it is to this reality, life, that the space-mind is related 
and in which its roots, its heart and the very center 
of its being are at one with the divine mind of the 
kosmos. 

The question of dimensionality is solely a concern 
of the objective or brain-mind which is the intellect. 
It is one of the ways in which the intellect endeavors 
to understand phenomena. It is an arbitrary con- 
trivance devised by the intellect for its convenience in 
studying the world of things. Without it, as ob- 
viously appears, the intellect would not be able to go 
very far in its consideration of the minor problems 
which inhere in material things. The fourth dimen- 
sion is but another attitude, another contrivance, which 
the intellect has devised in order that it may study 
from another angle the evanescent phenomena of the 
world of appearances. Having apparently exhausted 
the possibilities of motion in three dimensions, and 
being driven on to the acquirement of more picturesque 
views by the very necessity of its continued growth, it 
has betaken itself to another higher mountain peak, 
called "hyperspace" where with larger lenses and 
higher powered instruments it is beginning to scan 
the landscapes of a new intellectual realm of conscious- 
ness. Yet the celestial wonders of this new-found 
realm of consciousness remain in undisturbed forget- 
fulness or neglect. But it is not by a scrutiny of 



NORM OF SPACE DETERMINATIONS 201 

mathetic landscapes nor by a study of the celestial 
wonders that the Thinker shall one day realize the 
object of his eagerly pushed quest after the real; for 
he shall find it, if at all, in the temple of the kosmic 
mind which is not made by the intellect nor meted and 
bounded by geometric systems of space-measurement. 
In all the learned pother incident to the mastery 
of the phenomenal, the furniture of the world of the 
senses, it is as if the self in man, the Thinker, sat 
secluded in a six-walled tenement, and hence six times 
removed from the subject of his study, and endeavored 
to interpret that which appeared to his vision. And 
thus, thinking that what he sees is the only reality, 
he remains in inglorious nescience of the reality of 
that upon which he himself stands, unconscious that the 
tunnel-shaped aperture through which he peers leads not 
outward, but backward and within to the habitation 
of the real of which he himself is a part. Men are 
deeply and well-nigh hopelessly concerned with ap- 
pearances, with static views of life, with instantaneous 
exposures. Life, reality and all the eternal verities 
pass on and assume countless postures, attitudes, 
moods, tenses and nuances. The intellect is content 
to occupy itself with a single tense or mood. Indeed, 
it has no aptitude or power to consider more than one 
at a single time. It thus misses the continuity, the 
ceaselessness, of life. What is more, every singu- 
larity, every attitude, mood or tense which the intel- 
lect grasps for consideration is immediately remade 
so as to fit its own moods and tenses. And upon each 
and every nuance the intellect immediately imposes 
its own form — actually and literally rehabilitates them 
with its own habiliments. Unfortunately, this pecul- 



202 THE MYSTERY OF SPACE 

iarity occludes the intellect from any approach to the 
true nature of that phase which it can grasp. 

Hyperspace is one of the illusions of the phe- 
nomenal; it is the dress which the intellect has super- 
imposed upon a single nuance; it is a mask which is 
an exact replica of the mood of the intellect. Yet 
through this mask the intellect grandly hopes to ap- 
proach reality. The period through which the mind 
is now passing is a repetition of the evil days of 
scholasticism when men set out to determine the exact 
number of celestial beings that could be perched upon 
the extremity of a needle point. It is a time when 
men's minds easily assume grotesque and hideous 
shapes and their thoughts become the embodiment of 
fantastic entities. The exclusive occupation of such 
minds becomes the fabrication of mathetic monstrosi- 
ties which rapidly deliquesce upon the first approach 
of the real or the appearance of the first ray of intui- 
tion which may escape through the dim and misty con- 
dition of the intellectual over-hangings. It will not be 
ever thus; for the Thinker will one day pass from a 
study of the arrangement of phenomena in space and 
by well-ordered steps come once again to himself. 
And then through the maze of it all set out upon the 

true path the tridimensionalty of space following 

which he will inevitably approach the citadel of the 
real, the kosmic space-mind. 



CHAPTER VII 

The Genesis and Nature of Space 

Symbology of Mathematical Knowledge — Manifestation and 
Non-manifestation Defined — The Pyknon and Pyknosis — 
The Kosmic Engenderment of Space — On the Consubstan- 
tiality of Spatiality, Intellectuality, Materiality, Vitality 
and Kosmic Geometrism — Chaos-Theos-Kosmos — Chao- 
geny and Chaomorphogeny — N. Malebranche On God 
and the World — The Space-Mind — Space and Mind Are 
One — The Kosmic Pentoglyph. 

Geometry is concerned primarily with a study of 
the measurement of magnitudes in space. Three co- 
ordinates are necessary and sufficient for all of its 
determinations. Metageometry comprehends the 
study of the measurement of magnitudes in conceptual 
space. For its purposes four or w-coordinates are 
necessary and sufficient. Perceptual space is that form 
of extension in which the physical universe is recog- 
nized to have been created and in which it now exists. 
Conceptual space is an idealized conception belonging 
to the domain of mathesis and has no actual, physical 
existence outside of the mind. Mathematical space 
represents the idealism of perceptual space. 

Geometrical magnitudes may be defined as symbols 
of physical objects and geometry as a treatise on the 
symbology of forms in space. In fact, all cognitive 
processes are simply efforts at interpreting the sym- 

203 



204 THE MYSTERY OF SPACE 

bolism of sense-deliveries; and the difference between 
mere knowledge and wisdom, which is the essence of 
all knowledge, is the difference between the under- 
standing of a symbol and the comprehension of the 
essential nature of the thing symbolized. So long as 
knowledge of space is limited to the understanding of 
a symbol or symbols by which it is presented to the 
consciousness so long will it fall short of the compre- 
hension of the essential nature of space. In vain have 
we sought in times past to understand space by 
studying relations, positions and the characteristics of 
forms in space; in vain have we based our conclusions 
as to its real nature upon the fragmentary evidences 
which our senses present to our consciousnesses. It is 
as if one had busied himself with one of the meshes in 
a great net and confined his entire attention to what 
he found there, meanwhile remaining in complete ig- 
norance of the nature of the net, how it came to be 
there, of what it is made and how great its extent 
may be. 

There is ever a marked difference between a symbol 
and the thing which it symbolizes. Words are the 
symbols of ideas; ideas, as they exist in the mind, are 
the symbols of eternal verities as they exist in the con- 
sciousness of the Logos of the universe. There may 
be a wide diversity of symbolic forms which represent 
one single idea; as, for instance, the variety of word 
forms which represent the idea of deity in the various 
languages. Likewise there may be a multiplicity of 
ideas which represent a single verity. But neither is 
the idea nor the word the real thing in itself. That 
quality of a life-aspect which we call its thingness has 
an essential nature which cannot become the object 



GENESIS AND NATURE OF SPACE 205 

of consciousness except by virtue of its representa- 
tion through ideas and their symbolisms, and even 
then, the thing which we conceive is not the nature 
of a quality of the life-aspect but an idea of it — a 
symbol which stands for that idea. In order, there- 
fore, for the mind to arrive at an understanding of 
an eternal verity, such as space, it must first be able 
to synthesize all of the representative ideas and then 
abstract from their compositeness a notion of its 
essential nature. But this can be done only by identify- 
ing the consciousness with the essential being of the 
object considered. In other words, the consciousness 
and the intrinsic being of forms, principles, forces and 
processes must embrace each other in the intimacies 
of direct cognition; the life which is consciousness and 
that life which is essential being, being coeval, co- 
ordinate and mutually responsive, must in so close a 
contact as here intimated reach an understanding of 
the realism shared by both. That is, the human con- 
sciousness, following in the wake of life and consisting 
of a specialized aspect of life itself, will, by such an 
intimate approach to the life-principle of forms, 
readily understand; for it has only to recognize a 
replica of itself in rendering its judgment. But it is 
not claimed that such a state of recognition by the 
consciousness of life itself can be attained at all by 
ordinary means, neither is it believed that it is the 
next stage in conscious evolution. However, it is not 
doubted but that such an exaltation of the conscious- 
ness is possible, yea practical; but the difficulties which 
beset the path of attainment in this direction are so 
great that it may as well be considered unattainable. 
The mere fact of these difficulties, however, only re- 



206 THE MYSTERY OF SPACE 

emphasizes the insufficiency of the intellectual method. 
The identification of consciousness with essential being 
is a procedure which cannot be accomplished by an 
act of will directly and immediately. Because it is 
a process, a series of unfoldments, an adjustment of 
the focus of consciousness to the kosmic essentialities 
which constitute the substructure of the manifested 
universe. In the very nature of things, a kosmic 
essentiality cannot be viewed as being in manifestation 
especially in the same degree as ordinary physical 
objects are manifest. The former is a state, a poten- 
tiality, a dynamic force, an existence which should be 
thought of as an extra-kosmic affair dwelling on the 
plane of unity or kosmic origins; while the latter are 
the exact opposite of this. The one can be seen, felt 
and sensed while the other is the roots which are not 
seen but lie buried deeply in the heart of the uni- 
versal plasm of being and beyond the ken of sensuous 
apperception. The term manifestation is both rela- 
tive and flexible in its use. It is relative because it 
will apply equally to all stages of cognition. A thing 
is in manifestation when it is presentable to the 
ordinary means of cognition belonging to any stage 
of conscious functioning; it is not in manifestation when 
it is beyond the scope of the Thinker's schematism 
of cognitive powers. Its flexibility is seen in its ready 
yieldance to the entire range of implications inhering 
in the process of cognition, fitting the simplest as well 
as the highest and most complex. 

Great is the gulf which is interposed between 
manifestation and non-manifestation ; and yet the two, 
in essence, are one. They are linked together as the 
stem of a flower is joined to its roots. Likewise one 



GENESIS AND NATURE OF SPACE 207 

is visible, palpable while the other is invisible, im- 
palpable though no less real and abiding. As the thin 
crust of earth separates the stem, leaves and flowers 
from the roots so the limitations of man's conscious- 
ness separate the manifest from the unmanifest. 
Similarly, as when the surrounding earth is removed 
from the roots and they are laid bare revealing their 
continuity and unity with the outputting stem and 
flowers, so, when the limitations of consciousness are 
removed by the subtle process of expansion to which 
the consciousness is amenable so that it can encompass 
the erstwhile unmanifest, it, too, will reveal the eternal 
unity of the kosmic polars — manifestation and un- 
manifestation. 

There is but one barrier to ultimate knowledge and 
that is the human consciousness however paradoxical 
this may seem. The unutterable darkness which shuts 
out the so-called u unknowable" from our cognition 
is the limitation of man's upreaching consciousness. 
These limitations constitute the difference between the 
human intellect and the mind of the Logos. Never- 
theless the outlying frontiers of man's consciousness 
gradually are being pushed farther and farther with- 
out. Every new idea gained, each new emotion in- 
dulged, each new conception conquered, and every 
mental foray which the Thinker makes into the realm 
of the conceptual, every exploration into the abys- 
mal labyrinth of man's inner nature are the self's 
expeditionary forces which are gradually annihilating 
the frontier barriers of consciousness and thus ap- 
proaching more closely upon the Ultima Thule of 
man's spiritual possibilities. 

Space is in manifestation. It exists and has being 



208 THE MYSTERY OF SPACE 

whether it is viewed as an object, an entity or the 
mere possibility of motion. That it offers an oppor- 
tunity of motion and renders it possible for objects 
to move freely from point to point cannot be denied 
and yet this fact has no bearing whatever upon the 
essential nature of space. The very fact of its ap- 
pearance, its manifestation, makes it obvious that it 
is the nether pole of that eternal pair of opposites — 
manifestation and non-manifestation, being and non- 
being, which are essentially one. 

It will be seen from figure 17 that the period 
of involution embraces seven separate stages, the 
monopyknon, duopyknon, tripyknon, etc., being the 
unit principle or engendering elements of the respective 
stages. Involution comprises all creative activity from 
the first faint stirrings of the void and formless chaos 
until the universe has actually become manifest and 
dense physical matter has appeared. It is divided 
into two cardinal periods, namely, the chaogenic 
period during which primordial chaos is given its 
character and directive tendencies for the world age. 
It is a phase of duration wherein the fiat of kosmic 
order is promulgated, and consist of three stages, 
monopyknosis, duopyknosis, tripyknosis 1 gradually, 
insensibly, gradating into the fourth or quartopyk- 
notic; the chaomorphogenic period is likewise divided 
into three stages — quint opky no sis } sextopyknosis and 
septopyknosis, developing out of the fourth gradually. 
The quartopyknotic stage is the stage of meta- 
morphosis or transmutation wherein the transition 
from non-manifestation to manifestation is completed; 
it is also the stage of kosmic causation, because from 
1 See Fig. 18. 



GENESIS AND NATURE OF SPACE 209 

it spring the matured causes or "vital impetus" which 
engender all that follow. 



og 



I10H0PYKH0H 



DUOPYKNOH 




SEXTOPYKHOB 



SEHTIEHT 



SEPTOPYKKOH 



PHYSICAL 



Fig. 17. — Involution and Evolution 

The close of the involutionary phase of the world 
age is marked by the final deposition of dense physical 
matter and this is closely followed by the beginnings 



210 THE MYSTERY OF SPACE 

of the evolutionary movement which, like the involu- 
tionary movement, is divided into two cardinal periods, 
namely, the morphogenic (during which are pro- 
duced, in turn, insensible forms, sensible forms and 
spiritual forms) and the kathekotic period which 
marks the perfection, the consummation of the evolu- 
tionary movement. These two cardinal periods of 
the evolutionary phase of duration and the two cardinal 
periods of the involutionary phase complete the kosmic 
age, the "Great Day of Brahma." The concentric 
circles, beginning with the dot and ending with the 
seven concentric circles, and designated as a, b, c, d, 
e, f, g, are representations of the constitution of the 
respective units corresponding to each of the seven 
subdivisions. They symbolize the seven degrees of 
condensation or pyknosis which comprise the genesis 
of space, on the one hand, and on the other, the stages 
of unfoldment. Because, during involution all poten- 
cies, powers and characters were being infolded, in- 
volved; but during evolution, these are being unfolded, 
expressed, evolved. 

The figure 1 8 is another view of these two major 
movements, involution and evolution. The genesis of 
space is here shown symbolized by the Kosmic Egg. 
The seven stages of involution are referred to as, the 
monopyknotic, duopyknotic, tripyknotic, quartopyk- 
notic, quintopyknotic, sextopyknotic and the septopyk- 
notic; while the corresponding stages of evolution are 
referred to as, the physical, the sentient, mental, 
causative or spiritual, the triadic, duadfc and monadic, 
indicating that the principle of physicality is succeeded 
by the principles of sentience, mentality, spirituality, 
and the three forms of kathekotic being. This sym- 



GENESIS AND NATURE OF SPACE 211 

bolism, it should be stated, is designed with respect to 
the universe and man and has no reference to other 
possible evolutions than the human and contempora- 
neous animal, plant and mineral evolutions. 

To follow the ramifications of the symbolism 
above would involve a survey of all branches of 



Monopvknosis 



Duopyk 



Trtpyknosis 



Monacftc 



Quart opyknosi s 



Qumtopyknosis 



Sextopyknosis 



Septopyknosis 




Causative 



Sentient 



Physrcal 



Fig. 18.— The Genesis of Space 

knowledge, and indeed, would be out of place in this 
book. Only the widest general outlines can be sug- 
gested here and it is believed that this is sufficient to 
enable the reader to grasp the magnitude of the 
symbol and to understand its purpose and intent. 
It would appear, therefore, that if it is possible for 
the intellect to traverse by means of a study of kosmic 
symbolism, used as a standard of reference, the entire 



2i2 THE MYSTERY OF SPACE 

length of the bridge which engages the antipodes into 
an eternal unity, something may be gained in the way 
of a more definite and clearer understanding of the 
essential nature of space in its relation to kosmo- 
genesis. 

In the diagram, figure 17, is shown a table which 
represents the stages of space-genesis. It will be noted 
that the whole scheme is divided into seven stages. 
It is not an arbitrary division simply but a symbolic 
one and represents fullness, completeness, entirety. 
The names given, namely, "monopyknotic, duopyk- 
notic," etc., represent the symbolic characteristics of 
each stage in its relation to the universe in the process 
of becoming. The terms "monadic," "duadic," "tri- 
adic," etc., are representative of the seven planes of 
matter in the universe. 

A pyknon is a kosmic principle and represents the 
typal aspect of kosmogenesis. It is a generic term 
and may be identified in its relation to the various 
stages by the prefix. The monopyknon belongs to the 
ulterior pole of the antipodes on the side of non- 
manifestation. So do the duopyknon and the tripyk- 
non. Pyknosis is a process of kosmic condensation, 
or limitation for purposes of manifestation. It is a 
stage in the descent of the kosmic Spirit-Life, a degra- 
dation of non-manifestation into manifestation, and is, 
therefore, the cardinal causative principle of creation. 
The term pyknon being generic is applicable alike to 
a particle of matter, a state of being, a condition of 
existence, a process or a principle. The monopyknon 
is, accordingly, the primary aspect of the process of 
kosmic pyknosis. It is the archetype and therefore 
all inclusive and omnipotential. But whether regarded 



GENESIS AND NATURE OF SPACE 213 

as a singularity or as a whole it should never be 
divorced in thought from the primal act of creation. 
It represents the first act of material differentiation 
in the being of the creative Logos on the plane of 
non-manifestation. It is the beginning of every great 
planetary or kosmic manvantara or period of mani- 
festation. During either a planetary, solar or kosmic 
pralaya, or gestatory period, the kosmic plasm is it 
a quiescent, undifferentiated condition. This undif- 
ferentiated plasm when acted upon by the will of the 
Creative Logos, Fohat, as He is sometimes called 
in Eastern Philosophies, begins to become conditioned, 
begins to differentiate. The primal act or stage in 
such a process is the formation or appearance of a 
monopyknon. It then becomes the characteristic 
aspect of that stage. 

Monopyknons are the quiescent, unawakened, 
though potential and archetypal principles peculiar to 
the monopyknotic period of space-genesis which are 
ultimately to become, on the physical plane, singu- 
larities of life of whatsoever kind. Thus the lineage 
of every single life-form or principle in the universe 
runs unbroken back from the present of the Now to 
the present of monopyknosis. So great is the design 
of the kosmos that the entity which is now man or 
the atom was started on its journey to this culmina- 
tion at the break of the Great Kosmic Day when the 
omni-pregnant wheels of monopyknosis first began to 
turn. Duopyknons and tripyknons constitute the two 
remaining stages on the plane of non-manifestation. 
And their correspondences in every stage of involution 
or the descent of spirit into matter are eternal and 
kosmic. Likewise the lineage of the dual and triple 



214 THE MYSTERY OF SPACE 

aspects of all life forms on the path of evolution may 
be traced rearward to the duopyknotic and tripyknotic 
stages of kosmogenesis. 

The metamorphosis by which the monopyknon be- 
comes a duopyknon contrives the differentiation of the 
pristine plasm of kosmic being so that the first becomes 
the ensouling or vitalizing principle of the second; 
and, in turn, the second becomes the vitalizing or 
inner principle of the third; the third of the fourth 
and the fourth of the fifth and so on throughout the 
series until the last is reached which is the septo- 
pyknon. The septopyknon is, therefore, a seven- 
principled form. It is both unitary and septenary — 
unitary in the sense that the seven are really one and 
septenary for the reason that each of the seven 
principles, in the course of evolution, becomes a sepa- 
rate process specially adapted to functioning upon its 
peculiar plane of matter. Thus it is seen that the 
utmost significance attaches to this septenary pyknosis 
of the kosmic plasm of life. The implications of 
this conception are, of course, too vast and multi- 
farious to be set down here. We shall have to dismiss 
it with one observation only, and that is: every 
single appearance of life and form in the totality of 
such appearances is rooted in kosmic pyknosis where 
it has received its inner vitalizing force, its form and 
the law of its mode and manner of appearance to- 
gether with the metes and bounds of its existence. 

These processes, monopyknosis and duopyknosis, 
are to be regarded as taking place, each in its own 
period, everywhere throughout the Body of Being of 
the Logos but on the plane of non-manifestation. 
They are states of preparation for manifestation 



GENESIS AND NATURE OF SPACE 215 

analogous to the germinative period of the seed or 
the egg. They represent the first stirrings of the 
kosmic plasm and contain the promise and potency 
of all that is to succeed them. There is one other 
stage, coordinate with these two, and that is the 
tripyknotic which completes the unmanifest trinity and 
constitutes the archetypal vehicle whence proceeds the 
manifested universe. The ensouling principle of the 
tripyknon is the duopyknotic principle. But when the 
descent has reached the tripyknotic stage the three 
have been merged into one and the characteristics 
which were peculiar to each as a separate pyknon are 
then fused into a single quality having three aspects 
which are mutually interdependent and coordinate. 

The unmanifest trinity, now complete as a result 
of the triple pyknotic process, is the imperishable and 
ever sustaining radix, the all-mother of the manifested 
universe. It is the Golden Egg laid by the god Seb 
at the beginning of a great life-cycle. It is also 
Chaos-Theos-Kosmos , i.e., kosmic disorder, divine will 
or generating element, and kosmic order or space. 
In it, as in an egg, resides in kosmic potency all that 
the universe is to become in any "Great Day of 
Brahma" or any Great Kosmic Life Cycle. Its eldest 
born is space, physiological and perceptual, and the 
latter is the eternal father-mother of the universe. 
Space is, therefore, the male-female principle of 
manifestation; in its kosmic womb all forms are 
created, developed, evolved and sustained. Into it 
again, at the close of the Great Day, all existences, 
forms and all principles matured and ripened by the 
vicissitudes of kosmic evolution will be inhaled with 
the return of the Great Breath of Life. The un- 



216 THE MYSTERY OF SPACE 

manifest trinity is the archetype, and therefore, the 
pattern or model for the manifest, embodying in 
potency all that the manifest may ever become. Forth 
from the unmanifest proceeds space as a dynamic 
process endued with the potentiality for generating 
all that the manifested universe contains. 

The terms "unmanifest," "unconditioned" and 
"unlimited" have a special meaning here and are used 
in the same sense as the mathematical term "trans- 
finite," and therefore, imply a transcending of any 
finite or assignable degree or quality of manifestation. 
Hence they should be distinguished from the term 
"absolute" which has a different implication. So that 
although the triple process outlined above may not 
be viewed except as a characterization of the plane 
of non-manifestation, and hence of the primordial 
activity of the Creative Logos, there is nothing in the 
symbolism to warrant the identification of this process 
in any way with the Logos in absolution. For on this 
view Absolute Being is, in a large measure, sacrificed 
when the monopyknotic process is begun and the 
monopyknon (kosmic principle) begins to appear. 
Absolute Being, while it may not be defined, delineated 
or described may be symbolized by the ideograph: 
"action in inaction"; "being in non-being"; "mani- 
festation in non-manifestation" ; for these are symbols 
merely and do not describe or delineate. 

We have observed the subtle connection which 
exists between manifestation and non-manifestation 
and have seen how that, as the roots of a plant sustain 
the outer growth of stem and flowers, the former 
being the matrix out of which proceeds the latter, and 
that in like manner does the unmanifest sustain the 



GENESIS AND NATURE OF SPACE 217 

manifest; it should, therefore, be clear that the Body 
of Being of the Unmanifest Logos, in a similar 
manner, is the basis and cause of all that is manifest 
or in existence; and yet it is more, it is essentially all 
that is and all that the all is to be in manifestation at 
any time throughout the immeasurable process of 
kosmogenesis. It is the Self of the Universe, the 
meta-self of the great diversity of selves. The Self, 
however, although it cannot be said to exist except as 
a simple, homogeneous quantity is nevertheless, in the 
very nature of things, a triunity, and in essence, not 
only the basal element of the All-Space and the con- 
crete forms which exist therein, but is also identical 
in essence and substance with space. 

A stage has now been reached in the description 
of our symbolism when it may be assumed, upon the 
basis of the foregoing, that the meta-self has become 
manifest, i.e., in potential and dynamic appearance, 
as a result of the triple process of pyknosis herein- 
before outlined. The meta-self may then be identified 
with the Supreme Manifest Deity; for there is ever 
a subtle identification of the manifest with the unmani- 
fest. Out of their action and interaction the universe 
is made manifest and phenomenal and is thereby sus- 
tained. The reciprocity of action between these two 
kosmic polars, is the metamorphotic key to creation; 
it is the symbol of the procedure of Creative Will in 
the act of creating. It is the transitional process 
whereby the passage is made from non-being to being; 
from the unconditioned to the conditioned; from un- 
differentiation to differentiation and is reflected and 
symbolized in every natural process wherein matter 
is transformed from one state to another, or life and 



218 THE MYSTERY OF SPACE 

mind and spirit diffused, centralized and organized into 
ever new and higher forms and expressions. 

Another important notion to be gained in this 
connection is the fact that it appears as a logical 
sequence to the foregoing that the being of the mani- 
fested Logos must necessarily fill all space, yea more, 
is that space in every conceivable essentiality. His 
limitations are the limitations of space. His qualities, 
properties and attributes are the qualities, properties 
and attributes of space and are only different from the 
original spatial character when manifested through a 
diversity of forms by whose very inner constitution and 
exterior form the modifications are accomplished. All 
matter in the universe, all energy, and indeed, all 
manifestations or emanations of whatsoever character, 
hue, tone or quality are, in reality, His being and 
nothing but His being. There is, therefore, no form 
nor ensouling principle whether of life, mind or sheer 
dynamism which can exist outside of His being and 
be, even in the slightest degree, absolved from an 
eternal identity therewith. Once this idea is grasped 
and its varied implications noted it then is no longer 
conceivable that any other order or schematism can 
be possible in our universe, and that, too, despite the 
multiform conceptions peculiar to the varied systems 
of philosophy. 

The matutinal dawn of creation came at the close 
of the tripyknotic movement which resulted in the 
elaboration of materials, the initiation of principles, 
processes and types, and the final preparation of the 
field of evolution. The three processes or aspects of 
non-manifestation projected in preparation for mani- 
festation, namely, monopyknosis, duopyknosis and 



GENESIS AND NATURE OF SPACE 219 

tripyknosis represent the earliest stages of germinal 
development. When these had closed, the Great 
Kosmic Egg began to germinate; the first faint, in- 
describable signs of manifested life began to appear. 
Involution set in. The fourth or quartopyknotic 
stage, though only slightly differentiated, or rather 
representing that period of kosmic involution when 
that which is to become the manifested universe first 
begins to fall under the sway of kosmic order, is 
nevertheless the basis of all great world processes. It 
is just midway between the poles — manifestation and 
non-manifestation. During this stage the life elements 
are receiving the imprints of character, being endowed 
with directive tendencies and stored with such dynam- 
ism as will persist throughout the Great Life Cycle 
in which they are to manifest. It is here that begins 
the movement of involution, the storing away of those 
elements and factors, no more and no less, that are 
to show forth on the upward path of evolution; it is 
here that matter begins to assume form ; electrons, ions 
and atoms created, or, that those minute processes 
which on the evolutionary side are to culminate in 
these are originated. This is the metamorphotic 
stage. It is the laboratory of the universe wherein 
Fohat, the Creative Logos, prepares the materials 
out of which and in which the vast diversity of 
morphons or forms is created. Quartopyknosis, ac- 
cordingly, is the first active step, on the plane of 
manifestation, which results in the appearance of per- 
ceptual space and consequently of physical matter it- 
self as well as all the other grades of matter in the 
kosmos. Space, brought into existence by the act of 
the Creative Logos in imposing limitation upon His 



220 THE MYSTERY OF SPACE 

being, is in its primordial form composed of quarto- 
pyknons or quadruplicate principles and tendencies 
which act in unison and to the accomplishment of a 
single end or purpose. On this plane or during the 
continuance of this period of space-creation, the roots 
of universal law and order are produced. In it are 
planted the principles of good and evil and a sharp 
line of demarkation established between all the con- 
ceivable pairs of opposites which exist. It accounts 
for the duality of life and form. Male-female; 
father-mother; positive-negative; Rajah-Tamas (ac- 
tion-inaction) — all these find in this process their 
eternal origination. It is the stage of harmony, bliss, 
ideality, perfection, perfect equilibrium and balance. 
Here, innumerable ages before they actually appear, 
the glow-worm and the daisy, the amoeba and the 
dynosaur, man and the planetary gods alike abode 
their time awaiting the toppling of the scales of kosmic 
potency when all would be plunged headlong into the 
endless labyrinth of becoming. 

The quartopyknotic process is similar in all details 
to the three preceding processes, these latter being 
prototypes of all succeeding stages of involution. 
The quintopyknon, accordingly, symbolizes the quintu- 
plicative action of life in its descending movement 
toward the creation of matter in its densest form. 
That is, it is a five-fold principle acting in unison and 
kosmic consistency, infolding in the universal where- 
withal that which is to become mental matter on the 
side of evolution. Just, as may be seen in the dia- 
gram, Figure 17, the quartopyknotic process sym- 
bolizes, on the involutionary side of the life current 
that which is to become on the evolutionary side, 



GENESIS AND NATURE OF SPACE 221 

spiritual essence, so the quintopyknotic deposits the 
seeds of that which is to become mental matter. Dur- 
ing both the quartopyknotic and the quintopyknotic 
processes all the potencies and promises, residing on 
the plane of non-manifestation and destined to show 
forth as spirit and mind are brought into a fuller and 
more marked degree of manifestation and become the 
seeds of spirituality and mentality which are to ripen 
and be ladened with the fruitage thereof many ages 
hence. 

The reader should bear in mind that the processes 
here described are thought of as taking place at the 
beginning and as having their roots planted in eternal 
duration; that they refer to a period long before the 
universe even resembled anything like its present 
aspect; at a time even before there were individual 
minds to perceive it, before even the gods — solar, 
planetary, super-solar and super-planetary were in ex- 
istence to take part in the matutinal ceremonials of 
creation's vast hour of stillness. The mind must 
accustom itself to go back of appearances, back of 
time, back of space itself and discern the foundations 
of time, space and appearances being laid and to per- 
ceive that which is no less than the action of the 
Supreme Deity Himself in brooding over the primor- 
dial formlessness which is Himself, and from which 
will gradually evolve the universe of qualities, condi- 
tions and appearances. 

The quintopyknon is, therefore, the base of the 
mind-principle in the kosmos. All the qualities of 
mind whether in man or in the planetary gods, whether 
in the moneron or the tyrannosaur, in the mountain 



222 THE MYSTERY OF SPACE 

or the oak, reside in kosmic potency in the quinto- 
pyknon. 

Nicholas Malebranche, 2 in one of his very 
lucid moments, beheld the essential character of the 
symbology of space and was led to the conclusion that 
God is space itself. To him it was equally certain that 
all our ideas of space, geometrical or purely physio- 
logical, as well as our notions of the great supra- 
sensual domain of ideas, exist in the kosmic deiform, 
or body of the Logos of Being. He saw "all things 
in God." God did not create ideas; they are a part 
of God Himself; God did not create mind; it is a part 
of Himself; no kind of matter did He create; it is a 
veritable part of Himself and indissoluble from Him- 
self. The great outstanding implication of this 
philosophy is that our consciousness of God is but 
a part of God's consciousness of Himself; our con- 
sciousness of self and the not-self is but God's con- 
sciousness of these things. There is no existence of 
anything, either of the self or the not-self, except in 
this consciousness. It is refreshing, therefore, to note 
that although the approach is made from another and 
entirely different direction, almost the same conclusion 
as to the ultimate resolution of all chaogenetic ele- 
ments into what is the very systasis or consistence of 
the great kosmic deiform, is reached. 

But a marvelous vision comes with the dawn of 
this truth upon the lower mind. It establishes clearly 
the truth of Kant's notion when he said: "Since 
everything we conceive is conceived as being in space, 
there is nothing which comes before our mind from 

2 Vide Recherche, Chap. VII, also Philosophical Review, V. 15, 
p. 401. — Malebranche. 



GENESIS AND NATURE OF SPACE 223 

which the idea of space can be derived; it is equally 
present in the most rudimentary perception and the 
most complete." The mind cannot get away from the 
conception of space, because, out of the very essence 
of space, as a result of the quintopyknotic process, it 
was produced, created and organized. The idea of 
space is, therefore, not derived from things in space 
nor from their relations in space. It sprang up with 
self -consciousness. As soon as the Thinker became 
conscious of himself he became aware of space. The 
very state of self-consciousness implies space. The 
self in man is a specialized aspect of space. Indeed, 
it is a projection of space. The moment the self can 
say: "I am," it also can complete the declaration by 
saying: "I am Space!* When the self looks out from 
his six-walled cabin of imprisonment into the im- 
mensity of what we call space he looks out into that 
which is himself and his immensity; he perceives the 
source and the ever-present sustenance of his being 
and recognizes his identity therewith, provided he does 
not allow himself to become entangled in the philo- 
sophical difficulities which the intellect is prone to 
throw around the simple, yet marvelously complex, 
notion of self-consciousness. 

This should settle, once for all, the question of 
apriority. The a priori inheres in quintopyknosis or 
kosmic psychogenesis. It is the essential nature of 
mind; it is the mind's lines of organization; it is the 
law of the mind's being and action. All mental per- 
ception originates from things in space. No thought 
of any detail, of any state or condition, whether lim- 
ited or unlimited, related or isolated can be conceived 
except it be of things in space. And this is so, because 



224 THE MYSTERY OF SPACE 

mind and space are one. It is not so with our con- 
ception of time. Time is merely an aspect of con- 
sciousness in its limitation and does not inhere in the 
mind in the same manner that does space. In fact, 
it is not a part of the mind's nature as it has been 
shown that space is. It would, therefore, seem to be 
a grave mistake so to coordinate the two notions. 
Space is the progenitor of mind and is continually 
identified with mind. Time is the child of conscious- 
ness. That is, it is one of the illusions of conscious- 
ness which the ego will shed as his consciousness ex- 
pands. Duration alone is a coordinate of space. 

The mind now recognizes space as something apart 
and separate from itself only because of its uncon- 
sciousness of the identity existing between it and space. 
Just so, it is not by mind alone that the at-one state 
of consciousness shall be attained; for although in one 
form or another it is able to gain some knowledge of 
the apparent oneness of all life it cannot directly 
realize this oneness. In order to do this fully it must 
be able consciously to identify itself with the life, feel 
what it feels and experience what it experiences and 
otherwise come into a conscious relationship with the 
root and source of life. Space-consciousness is a 
simple, direct cognitive process; while time-conscious- 
ness is a complex, and therefore, indirect process. 
The former cannot be analyzed. That is, no analysis 
is necessary to its sufficient comprehension; the latter 
must always be analyzed and categorized for its suffi- 
cient apprehension. Every moment of time whether 
past, present or future, when presented to the con- 
sciousness, is determined by its relationship to some 
other moment of time. Space is indivisible; time is 



GENESIS AND NATURE OF SPACE 225 

divisible. Space is an intuitional concept; time is an 
intellectual concept. Time belongs to phenomena. 
Space is the root and source of phenomena. Time is 
the leaves of a tree while space is the life of the tree. 

Space-consciousness, in its relation to the present 
status of mind-development, is itself an illusion; for 
despite the fact that the Thinker's apperception of it 
as a state is simple, direct and fundamental, it is only 
so because of the inability to realize to itself the unity 
of the seeming two. The attainment of space-con- 
sciousness or the space mind, which contrives the un- 
derstanding of the identity of mind and space also 
annihilates the consciousness of space as a separate 
notion from the mind. Once the Thinker's conscious- 
ness has arisen to that state where it perceives its unity 
with space all sense of separateness is lost. Just as 
when two molecules of hydrogen uniting with one 
molecule of oxygen to form a new compound lose 
their identity in the new realization of unity, so does 
the consciousness when by the alchemy of psychogenesis 
it becomes identified with space, not only lose its 
identity as such, but also any consciousness whatever 
that space exists as something separate and distinct 
from itself. Imagine the whole of the duration aspect 
of kosmogenesis crowded into an infinitesimal instant 
and the bulk of all matter, suns, stars, worlds and 
planets, condensed into a space less than the magni- 
tude of an hydrogen ion and in this way a symbol of 
what it may mean to attain unto absolute knowledge 
or unto the space-mind, may be obtained. 

Recurring to the process of quintopyknosis, it may 
be noted that the quintopyknon or five-fold kosmic 
principle of life which we have seen to be identical 



226 THE MYSTERY OF SPACE 

to the seeds of mental matter brought into existence 
by the reaction of Fohatic energy or the Will of the 
Creative Logos upon the substance of the quartopyk- 
notic stage, is, symbolically speaking, more dense and 
compact than the pyknons of the preceding stages. It 
is ensouled by the quartopyknon. It is a rather com- 
plex state of ensoulment consisting of four condensa- 
tions or pyknoses. 

The next stage in the process of kosmic involution 
which is also concerned with the preparation of the 
evolutionary field is that of sextopyknosis and implies 
the senary condensation of the original world-stuff 
with the view to the formation of emotional or "astral" 
matter. Identically the same process of ensoulment 
or involution obtains upon the plane of sextopyknosis 
as have been observed to obtain upon the preceding 
planes of involution. Involution must necessarily pre- 
cede evolution. That which has not been involved, 
enfolded or ensouled cannot be evolved or unfolded. 
Whatever potencies, powers and capabilities or quali- 
ties and characteristics that may appear at any time 
in the universe of life and form must have first been 
involved or enfolded or else they could not have been 
evolved. Space itself is an evolution. It is a process 
of becoming, of unfolding, of flowering forth. As it 
evolves more and more there will appear new and 
added characteristics and qualities of life and form. 
New possibilities will arise and in the end a supernal 
vision of a glorified universe will burst into view. 

The scheme of space-genesis is completed during 
the septopyknotic process wherein the basal elements 
of dense physical matter and its various gradations 
are produced and given character, form and direction. 



GENESIS AND NATURE OF SPACE 227 

But this completion means merely a temporary estop- 
ment in the process of kosmogenesis which actually 
results in the formation of physical matter in its 
crassest state. It does not mean a final arrest of the 
entire process which is conceived of as continuing only 
in a regressive manner back to a kathekotic 3 condition 
wherein it embodies the fruitage of the entire scheme. 
The septopyknon, accordingly, is a seven-ply pyknon 
in which are embodied, in varying degrees of mani- 
festation and phanerobiogenic (life-exhibiting) quality, 
the essentialities of all that has preceded on all planes 
and during all stages of space-genesis. That is to 
say — in the physical life of the universe is confined the 
essence of all the series of grades of life in the kosmos. 
In man's physical body are wrapped up all the glories 
attainable in his long, almost unending pilgrimage 
of evolution; in it are stored all the possibilities of 
the spirit; all powers, all qualities, all characteristics, 
ever intended for man's attainment are in the physical. 
But they must be evolved, they must be unfolded and 
expressed. The physical must be glorified, spiritual- 
ized, deified. For by the way of the glorification and 
spiritualization of the flesh man may attain unto one- 
ness with the divinity in himself and consequently with 
the divine life of the world. 

To summarize: The genesis of space embraces 
seven stages, namely, the monopyknotic, the duopyk- 
notic and the tripyknotic which belong to the plane 
of non-manifestation and are the primordial world- 

*Kathekosis (from Chaos-Theos-Kosmos) is to evolution what 
"chaogeny" is to involution. It is the end of evolution, but also the 
beginning of involution, and in the latter function is known as 
"chaogeny." See diagram No. 17. 



228 THE MYSTERY OF SPACE 

stuff and together make up the unmanifest body of the 
Logos of Being. These become the seed-germ of the 
universe of spatiality. The quartopyknotic is the 
fourth stage in the process of space-genesis and is the 
metamorphotic or crucial stage during which non- 
manifestation is metamorphosed into manifestation. 
In it the unmanifest becomes the manifest. It cor- 
responds to the plane of pure spirit, and indeed, em- 
bodies within itself all the qualities which spirituality 
is to show forth during the life of the kosmos. The 
quintopyknotic is the fifth stage and corresponds to 
the mental plane, embodying in itself all qualities of 
mentality in the universe and furnishing the basis and 
essence of that which is to become the kosmic mind 
in manifestation. The sextopyknotic is the sixth 
process and symbolizes the sixth stage which embodies 
all the characteristics and properties of emotional mat- 
ter in the universe and is the basal element of the 
plastic essence of sentient existence in the kosmos. The 
septopyknotic is the seventh and final stage corre- 
sponding to the physical plane of the kosmos and 
contains in its seven-fold constitution the seeds and 
potencies of all the preceding stages, as well as all 
the characteristics and properties which physical matter 
is destined to show forth during the manvantara or 
world age. These seven processes result in the 
dynamic appearance of space, the mother and con- 
tainer of all things, and complete the involutionary 
aspects of kosmogenesis. Evolution began where in- 
volution ceased and will end for this manvantara when 
the last vestige of those powers, capabilities and 
potencies which were involved shall have been evolved 
unto kosmic perfection. 



GENESIS AND NATURE OF SPACE 229 

The measure of the Great Kosmic Space-form was 
sealed at the close of the involutionary movement of 
the Great Life Wave. Then its metes and bounds 
were fixed by the fringe of kathekosity which cir- 
cumscribed it. 

If it be true that the reader found it extremely 
difficult to grant the connotations of the symbolism 
when the mental or quintopyknotic stage was reached 
when illative cognizance was given to the fact that 
space is also composed of mental matter, it may be 
still more difficult to grant the claim that physical 
matter is also essentially a part of space. But this 
is the implication. And, therefore, it follows that all 
mattery all energy, all life and all mind wherever it 
may be found in the Great Space-Form is space itself 
and nothing hut space. Hence, it appears that space 
is indeed the dynamism of the universe. In its kosmic 
womb the great world egg was formed of its own 
substance solely and in it still the universe of form 
persists and evolves withal. 

If it be suitable for the physicist to talk of gravi- 
tation, electricity, magnetism and force let him do so, 
for these terms serve the present category of human 
knowledge ; but the human mind will not lament the day 
when it comes to recognize that these things, these forces, 
these aspects are nothing more than space-activities and 
space-phenomena. If a planet's place be preserved in 
space it is because space, vital, dynamic, creative space, 
sustains it and from its gentle, yet eternally firm grasp 
there is no escape. All that the planets, suns and 
worlds are and all that they may ever become in this 
manvantara or world age have been derived from 
space, yea, are of the very essence of spatiality. If the 



2 3 o THE MYSTERY OF SPACE 

chemist cKoose to talk of chemism, negative and posi- 
tive, of combining properties and dissociative phenom- 
ena let him also become aware that these phenomena 
are but the external aspects of the inner and ephemeral 
life-processes of space-forms and that ultimately these, 
too, may be traced back into an eternal originality 
within the bosom of the all-mother, spatiality. 

Dense physical matter, such as constitutes the 
physicality of celestial bodies in its ultimate dissocia- 
tion would, accordingly, be resolved into the original 
chaogenetic formlessness which marked the chaogeny 
of non-manifestation although it would naturally be 
orderly and progressive passing through the seven 
stages, septopyknosis, sextopyknosis, quintopyknosis, 
etc., until the end had been reached, meanwhile ex- 
hibiting in each plane the phenomena peculiar to the 
dissociative processes thereof. On this view space is 
a plenum of matter of varying degrees of intensity, 
ranging from the densest physical to the most tenuous 
and formless matter of the highest levels of the mani- 
fested universe. But as neither the dense material 
forms nor the other grades of matter have an eternally 
enduring quality, being alike subject to mutation, space 
likewise falls under the law of becoming whereby it, 
too, must yield to the edict of kosmic disorder. 

Some may be inclined to argue that since space and 
mind are one and the same thing it must necessarily 
follow that whatever possibilities of measurement may 
be found to exist in the mind would logically be found 
to exist in space; and that since all the necessary con- 
ditions of hyperspatial operations are proved to be 
existent in the mind the case of the hyperspatiality of 
perceptual space is proved thereby. In other words, 



GENESIS AND NATURE OF SPACE 231 

if the fourth dimension can be proved to be mentally 
construable it is also possible in perceptual space. But 
these hypotheses are not granted, and neither will they 
be acceptable to those minds who choose to take that 
view when it is known that there is a marked difference 
between the mind that is purely intellectual and mind 
that is purely intuitional or mind a priori. The in- 
tellect is fashioned for matter only; it is so constructed 
as to fit squarely into every nook and cranny, every 
groove and interstice in matter; yet for the generating 
element, life, it has no aptitude nor suitable con- 
gruence. 

The attainment of the space-mind or kosmic con- 
sciousness would then imply a mastery of all funda- 
mental possibilities pertaining to all degrees of matter. 
Thus by becoming conscious in the matter of all the 
planes one makes a certain definite approach to this 
ultimate state of consciousness until all the barriers 
between ordinary self-consciousness and the conscious- 
ness of the space-mind have been entirely obliterated. 

Pyknosis, in all of its septenary aspects, is con- 
cerned primarily with involution or the preparation of 
the chaogenetic elements for the work of kosmic 
evolution. It may be thought of as being divided into 
two great divisions, namely: chaogeny and chao- 
morphogeny. It is concerned with the organization 
of chaos, the establishment of kosmic geometrism in 
the formless, void, arupic substance and preparation 
for evolution. Chaogeny, of course, is that kosmic 
process by virtue of which space itself becomes mani- 
fest and in which there is no established order. 
Chaomorphogeny (from Chaos+Morphe+Geny) sig- 
nifies the activities of the creative Logos in laying the 



232 THE MYSTERY OF SPACE 

foundations in primordial space-matter of the various 
star-forms, including nebulas, worlds, planets, suns, 
etc., of which Canopus, Jupiter, Fomalhaut and Sirius 
and our own sun are examples, giving direction and 
general tendence to their varied life-processes. Both 
these processes are concerned with the preparation of 
the field and its consequent fertilization in anticipation 
of its cultivation and harvest. These two constitute 
kosmic involution or the great life wave's passage on 
the downward arc of the Great World Egg or Circle. 
It is during the chaomorphogenic cycle that the con- 
stitution of the universe of manifestation is promul- 
gated; when laws for its government during that 
manvantara are sketched out in the world of nascent 
spatiality; when the archetype of every imaginable or 
possible form is projected upon the impregnated 
screen of creation, then folded in, pushed toward the 
center, involved, awaiting that time when the life wave 
begins its passage upon the upward arc and evolution 
ensues, calling forth all that has been enfolded in the 
bosom of the pyknotic centers of manifestation. It is 
easily conceivable that here during the troublous times 
of the chaomorphogenetic enfoldment the now known 
six directions of space were among the eternal edicts 
of space-genesis and that that law which now makes 
it appear that three coordinates, and only three, are 
sufficient for the determination of a point position in 
space was imprinted in the very nature of that which 
was to become space. 

The kosmic field having been prepared as a result 
of the chaomorphogenic activities, lowly and scarcely 
organized forms begin to appear and the ascent upon 
the upward arc of the Great Cycle commences. Evo- 



GENESIS AND NATURE OF SPACE 233 1 

iution begins. Its scope is likewise divided into two 
great stages, namely: (a) Morphogeny, the purpose 
of which is the development of life-forms or pyknons 
which are to appear on the various planets, stars, 
worlds and suns of the universe. It embraces the 
whole span of the life-aspect on the evolutionary side 
of manifestation. In this aspect is included also every 
conceivable adaptation of the universal principle of 
life from the beginning of its movement to the end. 
The universe is now functioning in and progressing 
through the vicissitudes of this stage. That is, all the 
present observable adaptations which the life-pyknon 
or principle is making, has made or will make, are 
embraced within the scope of what is here designated 
as the morphogenetic aspect of evolution, (b) The 
second and last division of the arc of evolution is called 
Kathekos, thus symbolizing the syncretism of the 
trinitarian aspects of kosmogenesis, Chaos-Theos- 
Kosmos, perfected and united as a result of the labors 
of manifestation. In this final summation of the labors 
of the life-wave as it has progressed from involution 
through all the devious manifestations of evolution are 
embodied the perfection and ultimate elaboration and 
expression of all the pyknotic tendencies which were 
established during the entire scope of space-genesis. 

Thus it will be seen that the first three stages of 
space-genesis, called chaogeny, encompass the first three 
pyknotic processes or are analogous thereto while the 
latter called chaomorphogeny, the organization and 
ensoulment of space-forms, embraces the latter four, 
quartopyknosis, quintopyknosis, sextopyknosis and sep- 
topyknosis. This division obtains on the involutionary 
side of the great life-cycle. The upward arc of the 



234 THE MYSTERY OF SPACE 

Great Kosmic Egg or Cycle is also divided into two 
great stages, namely, Morphogeny (manifestation of 
life through the various forms which it assumes) and 
Kathekos, or the kathekotic plane of perfected triunity 
which is represented by the evolutionary union of 
Chaos-Theos-Kosmos. Kathekos would, therefore, 
symbolize the ultimate elaboration of Chaos into a 
well-ordered kosmos wherein are expressed all the 
possibilities which inhered in the archetypal plan of 
the Creative Logos or Theos and in which all had 
reached the ultimate perfection in the body of being 
of the Logos Himself. But the kathekotic plane is 
to be distinguished from the original Chaos-Theos- 
Kosmos represented as functionating upon the plane of 
non-manifestation during chaogeny. Kathekos sym- 
bolizes the perfected manifestations of the triune 
aspects of the Creative Logos through the perfected 
forms resulting from the labors of kosmic evolution, 
while Chaos-Theos-Kosmos symbolized, as a. triune 
glyph, the Unmanifest Trinity in the primordial be- 
ginnings of space-genesis. One is the seed; the other 
the fully matured plant; one the egg; while the other 
is the full grown bird; one the root; the other the 
fruitage; one Alpha, the other Omega; one the be- 
ginning, the other the end. The end, however, is 
reached only that, in due time, the entire scheme may 
be commenced again, once more utilizing the results 
of the preceding scheme of evolution as the basis 
of the ensuing one. Thus after every Kosmic Day, 
commences the Kosmic Night. The succession of 
kosmic days and nights is infinite. This infinity of 
becomings in the life of the kosmos is a necessary 
outcome of eternal duration. 



GENESIS AND NATURE OF SPACE 235 

The above, thus briefly set down, is the symbolism 
of space-genesis. It is commended to the reader as a 
basis for the conception that the real, essential, per- 
ceptual space is something far more wonderful, more 
fundamental than either the geometrician or the meta- 
geometrician has ever dreamed of, and yet the latter's 
consciousness is undoubtedly being appulsed by the 
fingers of a new species of conceptualizations which, 
one day in the not too distant future, will arouse in 
it the faint hungerings after the realization of the real 
space-nature. These mathetic appetites thus brought 
into being will finally lead the human mind into the 
Elysian fields of kosmic consciousness where for an- 
other million years, perhaps, it may feed upon the 
mysteries and hypermysteries to be found in the 
granaries of the Space-Mind. 

The study of space in its wider and deeper mean- 
ings is necessary in order that a clearer understanding 
of its true significance, as the subject of geometric 
researches, may be gained. It is confessed, however, 
that there is neither direct evidence nor implicative 
authority for any assumption that the view herein out- 
lined affords any justification for the notion of the 
^-dimensionality of space. For, although the line of 
reasoning indulged in must lead inevitably to the con- 
clusion that the worlds of spatiality, materiality, in- 
tellectuality and spirituality, essentially and funda- 
mentally one so far as origins and qualities are con- 
cerned, were alike engendered by the same generating 
element, life; and that spatiality being the primal basis 
of the others is, nevertheless, under the exigencies of 
this aspect of the kosmos, highly susceptible to the 
mensurative requirements of the grossest, there ap- 



236 'THE MYSTERY OF SPACE 

pears to be no necessity for calling upon extraneous 
considerations for assistance in our efforts to com- 
prehend the various connotations of the symbol- 
ism. Then, too, it is easily conceivable that under 
conditions where these elements, spatiality, intellectu- 
ality and materiality, are not only co-extensive but 
interpenetrative, there is no justification for the as- 
sumption that they must exist in layers or manifold- 
nesses or in discrete degrees, separated from one 
another as if they were constituted of different sub- 
stances and occupied different spheres. For every 
single point in perceptual space is a focus for lines 
drawn through every conceivable grade of materiality, 
spatiality or intellectuality in the kosmos. And the 
same system of coordinates which is necessary and 
sufficient for the localization of a point in our space 
is also sufficient for the location of a point anywhere 
in the entire world of spatiality, intellectuality or 
spirituality. In fact, the external, visible worlds of 
materiality and spatiality are nothing more than the 
mass-termini of lines extending from divinity to 
physicality; from primordial originality to kosmic 
modernity and it is intellectually conceivable that pro- 
gression back over the grooves made by these mass- 
termini of lines would lead directly and unerringly to 
originality itself. In spite of the manifold pyknoses 
which we have shown to characterize the symbolism 
of space-genesis it is a very simple matter; for the 
entire scheme could and must have proceeded along 
strictly tridimensional lines. Tridimensionality must 
have inhered in the primeval archetype of space or else 
it could not appear as an outstanding fact of perceptual 
space now; for all that we can now observe in space 



GENESIS AND NATURE OF SPACE 237 

as characteristics must have first been included, en- 
folded, involved, before it could have been evolved. 
Hence, it is to be remembered that we are to-day deal- 
ing with the expressions of tendencies and principles 
which inhered in the manifested universe as poten- 
tialities in the very beginning. 

The alphabet of space-genesis consists of five char- 
acters, namely, the point, the line, the triangle, the 
square and the circle. These are the pentagrammaton 
of space, of intellectuality, materiality and of spiritu- 
ality. They constitute the basis of kosmic geometrism. 
With these all geometrical figures may be constructed; 
with them all magnitudes may be delineated and pro- 
jected. They describe every conceivable activity of the 
Creative Logos and designate the bounds of the entire 
scope of motility of kosmogenesis. 

In figure 19, are shown the dot, the line, triangle, 
square and the circle which together form the kosmic 
pentoglyph. The point symbolizes kosmic inertia, 
inactivity or the beginning of motion; the line 
is the first aspect of motion, the beginning of creation; 
the triple aspect of kosmogenesis is symbolized by 
the triangle, Chaos-Theos-Kosmos, the Unmanifest 
Trinity; the square emblematizes kosmic being in evo- 
lution; while the circle is the syncretism of all these 
and stands for the perfected kosmos, or the kosmos 
in process of perfection. Very truly did Plato re- 
mark: "God geometrizes" ; for the pentagrammaton — 
the point-line-triangle-square-circle — is the deity's way 
of manifesting Himself. But there is here no need 
for space-curvature nor for triangles whose value is 
greater or less than 180 degrees; there is no need even 
for the mathematical fourth dimension. 



238 



THE MYSTERY OF SPACE 



It cannot be believed, however, that metageometri- 
cians are really in earnest in what they suggest of 
hyperspace and ^-dimensionality; it cannot be believed 
that they are entirely satisfied with what they have 
found of the so-called hyperspatial, and yet, some of 
them are fanatically patriotic over the new-found 
domain; some are even intolerant. But there are 




Fig. 19. — Kosmic Pentoglyph 



others who look upon the fabric of metageometry as 
a stepping stone to space-realities, a mile-post on the 
path to the realization of a higher consciousness, the 
consciousness of the space-mind or kosmic conscious- 
ness. And may this not, after all, be the goal of the 
human intellect, now slightly distraught by the ex- 
uberances of youth and the joys of a new mental free- 
dom? The work of the future mathematicians will 
be the destruction of the tumorous inconsistences to 



GENESIS AND NATURE OF SPACE 239 

be found in the various non-Euclidean systems of 
geometrical thought, the elimination of the novelties 
and the nonsensicals, the synthesizing of those elements 
which are sanctioned by the space-mind and the build- 
ing thereon a sane interpretation of space-phenomena 
in the light of illuminations received from greatly ex- 
tended faculties and a participation in that larger con- 
sciousness into which the human race seems slowly 
to be immerging. 

The domain of hyperspace is but the fairy-land of 
mathesis, peopled with goblins, gnomes, kobolds, elves 
and fays which are the spaces, dimensions, proposi- 
tions, ensembles and theorems of the metageometri- 
cian. But like the fairies and nature spirits of the 
unseen about us, they have their bases in the real, 
objective world of facts however difficult it may be 
to establish their direct connection with it. As the 
gay, invisible sprites of phantom-land represent intel- 
ligent natural forces at work in the furtherance of 
the evolution of forms, so the impalpable things of 
mathesis are emblems of kosmic forces at work in the 
upbuilding of structures of higher consciousness which 
shall be towers of vision for the human soul whence 
it may view the hill-crests of infinite knowledge and 
the low-lying plains of kosmic mysteries. 

Finally, it has been noted that space is the very 
consistence of the kosmos; it is the life, the form and 
both the outer and the inner manifestation of the com- 
bined life and form; it is reality, also illusion; it is 
concrete, also ideal. We have noted also that mind 
is consubstantial with space and that space gives it 
its inner life and nature as well as nourishes its outer 
growth and development. In fact, we have seen that 



240 THE MYSTERY OF SPACE 

space and mind are one essentially and that they exist 
as aspects of the same thing, life. In whatever way, 
then, that the mind normally views space that is the 
natural way. All attempts to deviate from the natural 
way are, therefore, unsanctioned by the nature of 
things. So long as geometry remains true to the 
nature of mind and space so long will it be valid uni- 
versally and possessed of kosmic necessity and in- 
variance. It behaves most unseemly when it departs 
from its fealty to the nature of things per se. 

Both the outlook of the mind upon the objective 
world as well as its inlook upon its own states of 
consciousness or the subjective world are tridimen- 
sional. Its growth is tridimensional, its nature is 
likewise tridimensional, and there is not even the 
slightest tendence either to perceive, conceive or per- 
form in a four-dimensional manner, mathematically 
speaking. Trace out the biologic development of each 
mental faculty, from the mind of the moneron to the 
mind of the most highly developed man and it will 
be found that everywhere and always, without varia- 
tion or exception, the nature of each of these has been 
to express itself tridimensionally and naturally. There 
is not even the slightest sign of so much as a germinal 
appetence for the four-space; it would, therefore, 
seem almost a prostitution of mental faculty to divert 
mental energy into the seemingly useless channel of 
present-day metageometrical researches; yet, it must 
be admitted that even though the end sought cannot 
be attained, the final results of the intellectual delvings 
into the dread homogeneity of kosmic origins and the 
consequent realization of the awesome coevalism of 
mind and space whence shall arise the recognition of 



GENESIS AND NATURE OF SPACE 241 

the wondrous unitariness of all existences, will be that 
we shall come upon that thrice mysterious contrivance 
— the Heart of Divinity, the Kosmic Space-Center in 
which abide the roots of the Great All in a marvel- 
ously indescribable unity and infinite originality. 

We may conclude, then, that hyperspatiality and 
all its appurtenances are but the toys of the childhood 
of humanity. But, as the years pass and the days of 
maturity come on apace, it, toy-like, will also be dis- 
carded. And the mind will seize then upon the seri- 
ousness of reality just as the matured youth responds 
to the stern realities of life and manhood responsi- 
bilities. But no one can say that the toys of childhood 
are wholly useless; no one can say that the joys which 
they bring are entirely fatuous and unreal nor shall 
we attempt to intimate that mathetic contrivances are 
without utility, without purpose and significance in the 
life of the growing mind of humanity. But they, too, 
will pass away. 



CHAPTER VIII 

The Mystery of Space 

The Thinker and the Ego — Increscent Automatism of the In- 
tellect — The Egopsyche and the Omnipsyche — Kosmic 
Order or Geometrism — Life as Engendering Element — 
The Mystery of Space Stated — Kathekos and Kathekotic 
Consciousness — Function of the Ideal — The Path of Search 
for an Understanding of the Nature and Extent of Space 
Must Proceed in an Inverse Direction. 

The fragmentariness of the Thinker's outlook 
upon the universe of spatiality is due to the inhibitive 
action set up by the constrictive bonds which his com- 
plicate mechanism of intellectuality interposes between 
himself and reality. The Thinker, who stands back 
of and uses the various media of objective conscious- 
ness, such as the neural mechanisms, brain, emotions, 
his individualized life-force and the mind which to- 
gether make up the instruments with which he con- 
tacts the sensuous domain, by adapting his conscious- 
ness to these means, as the artisan utilizes his tools, 
constitutes his own intellectuality. The intellectuality, 
then, is the totality of media by which consciousness 
effects its entrance into the sensuous world and by 
which it receives impressions therefrom. In other 
words, it is the sum of all those qualities, operations, 
processes and mechanisms which are recognized as 
constituting the modus vivendi of man's intellectuality, 

242 



THE MYSTERY OF SPACE 243 

and these are, in reality, nothing more than the ego 
himself. 

Many have been inclined to regard that which has 
been called the ego as the highest sovereign power in 
the state of manhood. He has been looked upon as 
the final consideration in the constitution of the human 
being. But the ego is an evolutionary product and 
the concomitant of self-consciousness which is the 
I-making faculty in man's psychic life. It is that 
quality of consciousness which makes man conceive of 
himself as a separate, detached and independent be- 
ing. It is a purely intellectual or tuitional product, 
and, as such, is to be differentiated from the intuitional 
or life-quality which is the essence of man's real self- 
hood. With respect to the Thinker, the ego occupies 
precisely the same status as the agent to his principal. 
As the agent is the representative of the principal in 
all matters which come within the scope of his pre- 
scribed jurisdiction so is the ego the agent of the 
Thinker who is a spiritual intelligence. Accordingly, 
from an ethical viewpoint the Thinker is responsible 
for the acts of the agent and can in no wise escape the 
penalties accruing as a result of the agent's violations. 
Just as a commercial firm sends out a representative 
for the collection of data concerning certain phases 
of its business or it may be of any business or the 
entire world market so the Thinker projects his own 
consciousness into the mechanisms which are in their 
totality the egoic life. That is, he sends out his agent, 
the ego, into life and into the objective world of facts 
and demands that he shall convey to him, from all 
points of the territory which he is expected to cover, 
reports of his findings. Of course, these reports which 



244 THE MYSTERY OF SPACE 

are transmitted by the ego (the intellectual mechanism 
of the Thinker) are more or less well prepared sum- 
mations of his individual observations and deductions. 
These are the percepts which the ego presents to the 
Thinker's consciousness. Concepts are formed by the 
Thinker in his treatment of these sense-presentations. 
It very frequently happens that the ego transmits re- 
ports which, for one reason or another, give very 
imperfect knowledge of the matter which his reports 
are designed to cover. Often it is necessary that addi- 
tional and supplemental reports be made about the 
same thing, and even then, it is well-nigh impossible, 
if not quite so, for him fully to cover every detail of 
the matter under consideration and in no case is it 
possible for him to do more than report on the super- 
ficialities of the question under scrutiny. If the ego, 
in his operations, be imagined to be hampered by 
similar circumstances and difficulties as those which 
would ordinarily beset a commercial attache it will 
then be clear that his reports must ever be fragmentary 
because of the inaccessibility of much of the data which 
would be necessary for a full report, and further, be- 
cause of the inadequacy of his methods and means of 
gathering data due to the inherent limitations of his 
capabilities, endurance and perspicacity and innumer- 
able other limitations and difficulties which must be 
faced in all search for the real. So that, while the 
sufficiency of the means which the ego enjoys at this 
stage for all practical purposes is granted no hesi- 
tancy is entertained when it comes to a discovery of 
the reals of knowledge in declaring their insufficiency. 
Then, too, when it is remembered that these egoic 
reports are in the nature of neurographical communi- 



THE MYSTERY OF SPACE 245 

canons which are similar to telegraphic despatches and 
must pass through several stations, as ganglia, etc., 
often being relayed from one to another, it will be 
quite apparent that much, even of the original quality 
of the missives forwarded, will have been lost or 
radically changed in some way before it is finally de- 
livered for the inspection of the Thinker himself. It 
not infrequently happens, even in perfectly normal 
beings, that the ego in filing, recording, transcribing, 
interpreting, translating and otherwise preparing these 
data for the Thinker's use, lets a cog slip, misplaces 
some of the data, loses or destroys fragments of it 
and so is unable to maintain a complete portfolio of 
his materials. 

As the Thinker is entirely dependent upon his 
agent, the ego, for the trustworthiness of his informa- 
tion covering the matter of the sensuous world it is 
obvious that at best his information is very frag- 
mentary indeed, and necessarily so when it is consid- 
ered that the modus operandi of his agent and the 
difficulty of his operations are so complicate as to 
magnify the obstructions in the way to complete free- 
dom in this regard. 

To continue the similitude of principal and agent 
it may be asserted that it is also true that the com- 
mercial house that sends out its attaches frequently 
will send letters containing directions as to procedure, 
sometimes censuring for past delinquencies and some- 
times commending for praiseworthy deeds; and this, 
too, in addition to the original instructions which were 
given at the outset. It even comes to pass that the 
home office, because of some meretricious accomplish- 
ment, as the marked increase of efficiency shown by 



246 THE MYSTERY OF SPACE 

the agent's close application to his duties and the con- 
sequent success of his operations, confers certain favors 
upon the agent or removes some of the restrictions 
which were originally imposed, gives an increase in 
salary or promotes the agent to a higher and more 
lucrative office with larger powers and greater author- 
ity. This is analogous to what the Thinker does for 
the ego. For he not only receives reports from the 
ego, but often, in the shape of intuitions, gives addi- 
tional information as to the proper manner of doing 
things, sheds more light upon some obscure opera- 
tion, commends for duty well performed, condemns 
for failures or for wrong-doing, rewards arduous toil 
with greater powers of vision, keener insight, greater 
capabilities; in fact, promotes the ego in the sight of 
other egos by marking him out as an exceptional ego. 
But the curious aspect of this procedure is that, in time 
and after the ego has been repeatedly commended 
and promoted and otherwise favored by the Thinker, 
he begins to think that he owns the firm, that he is 
the life and main support of the whole corporation. 
He becomes arrogant, self-willed and finally falls into 
the illusion that he alone is responsible for the phe- 
nomenal success of the firm. This is the source of that 
illusion of the intellect which makes itself think that it, 
the ego, is all there is to man, that his instruments of 
operation in the objective world are the only kind 
of instruments that may be used; that his method of 
gathering data about things is the only safe and sure 
method; and so it develops that the intellectuality is 
the source of man's separateness, his individuality and 
his apparent aloofness from other men and things. It 
is, of course, needless to point out that in this way 



THE MYSTERY OF SPACE 247 

the intellect comes to be the tyrant of man, ruling with 
a rigid monopoly and as an all-exclusive autocracy. 

From the above implications it would appear that 
the intellect and the intuitive faculty are two separate 
and distinct processes, and so they are. One is the 
inverse of the other. The tendence of the egoic life 
or the intellect is for the external while the intuition 
is an internal process. The intellect acts from without 
towards the interior while the intuition acts from 
within outward. The intellect is the product of the 
intuition which is another term for the consciousness 
of the Thinker on his own plane. Just as the child 
lives a separate and distinct, though dependent, life 
from the parents so the intellect has a modus vivendi 
which is distinct and separate from that of the Thinker, 
and yet it is in all points dependent upon the life of 
the Thinker. Here again, we find an analogy in the 
relation of the child to the parent. As some children 
are more amenable to the will of the parent than 
others, so, in some persons, the intellect is more amen- 
able to the action of the intuition than in others. Yet 
it is a certain fact that the more the outward life is 
governed by the intuition, i.e., the more the intellect 
responds to the intuitive faculty of the Thinker, the 
higher the order of the life of the ego and the more 
accurate his decisions and judgments. In fact, it as- 
suredly may be asserted that the place of every in- 
dividual in the scale of evolution is determined in a 
very large measure by the degree of agreement be- 
tween the intuition and the intellect or by the ease with 
which the intuition may operate through the intellect 
as a medium. At least, the quality of one's life may 
be determined directly by these considerations. 



248 THE MYSTERY OF SPACE 

The Thinker being himself a pure spiritual intel- 
ligence, living upon the plane of spirit and therefore 
unhampered by the difficulties which the ego meets in 
his operations in the objective sensorium, and pos- 
sessed of far greater knowledge, is correspondingly 
free from the limitations of the ego and very naturally 
closer to kosmic realities. Hence, he is better situated 
for the procurement of correct notions of relations, 
essentialities and the like. It is believed, therefore, 
that in the proportion that these two processes, the 
intellectual and the intuitional, are brought, in the 
course of evolution, to a closer and more rigid agree- 
ment, in the proportion that the Thinker is able to 
transmit the intuitograms in the shape of concepts or 
that the intuition is made more and more conceptual, 
in just that proportion is humanity becoming perfect 
and its evolution complete. The difficulty found to 
inhere in the conceptualization of intuitions so that 
they may be propagated from man to man seems not 
to lie in the Thinker himself, but more essentially in 
the ego, in the intellectuality and its complicate 
schematism or plan of action. It would appear, 
therefore, that the only way of escaping or transcend- 
ing this difficulty is for the ego so to refine his vehicles 
or so facilitate his plan of action by eliminating the 
numerous relays or sub-stations intervening between 
the consciousness of the Thinker and that which may 
be said to be his own that the transmission of in- 
tuitograms may be accomplished with the greatest ease 
and clearness. While no attempt will be made to in- 
dicate the probable line of action which the ego or 
objective man will adopt for this purpose, it is believed 
that it may be said without pedanticism that the only 



THE MYSTERY OF SPACE 249 

true method of attaining unto this much desired state 
of things is, first of all, by assuming a sympathetic 
attitude not only towards the question of the intuition 
itself but to all phenomena which are an outgrowth 
of, or incident to, th© manifestations of the intuitive 
faculty through the intellectuality, and second, by the 
practice of prolonged abstract thought, this latter pro- 
cedure effecting a suspension of the intellectuality 
temporarily at the same time allowing it to experience 
an undisturbed contact with the intuitional conscious- 
ness, thereby laying the basis for future recognition 
of its nature and quality. It would seem that these 
two conditions are absolutely necessary in order that 
a more congruent relationship may be promoted be- 
tween these two cognitive faculties. Ordinarily, it 
would appear that the philosopher who is undoubtedly 
inured to the necessities of continuous abstraction or 
the mathematician whose most common tasks naturally 
fall in this category would be among all men most 
apt to develop to the point of conceptualizing intuito- 
grams readily, yet it seems that this is not the case. 
And there is good reason for it. The mind of the 
philosopher and the mathematician is intellectual 
rather than intuitional and is, therefore, wedded to 
matter, to the action and reaction of matter against 
matter and hence operating in a direction at variance 
with the trend of an intuitional mind. And this con- 
dition is undoubtedly due to a lack of a sympathetic 
attitude towards this species of consciousness. At any 
rate, it is thought that a too great anxiety in this 
respect need not be entertained by humanity at all, 
for the reason that in the case of a faculty, the rudi- 
mentary outcroppings of which are so marked and 



250 THE MYSTERY OF SPACE 

universally observable and existing in greater or lesser 
degrees in various human beings, there is ample evi- 
dence for the belief that it is being carefully and duly 
promoted by a well-directed evolution of psychic 
faculties and powers, so that at the proper time, de- 
terminable by the state of perfection reached by the 
intellectuality or the ego in the operation of his cog- 
nitive processes, the much desired agreement of these 
two faculties will have been realized and the concep- 
tualization of intuitograms into propagable conceptions 
an accomplished fact. Until this goal shall have been 
reached and the intuition shall have overshadowed the 
intellect as the intellect now overshadows the intuition; 
or the consciousness of the ego, derived from the in- 
terplay of the Thinker's consciousness among the 
various elements which constitute the ego himself, shall 
have been merged with that of the Thinker, the out- 
look must remain fragmentary, only becoming a well- 
ordered whole as the barriers of dissidence are broken 
down in succession. 

The evolution of consciousness, from the simple, 
undifferentiated moneron to the differentiated cell and 
from that to the cell-colony and from the cell-colony 
to the organism, traversing in successive paces 
through all the stages of lower life — mineral, vege- 
table and animal — to the stages of the simple, com- 
munal consciousness of the higher animals, to the 
self or individual consciousness of the human being, 
each requiring millions of years for its perfection be- 
fore a more advanced stage is entered, has been one 
continuous relinquishment of the lower and less com- 
plicate for the higher and more complex expression 
of itself tkrough the given media. When a newer and 



THE MYSTERY OF SPACE 251 

higher stage of consciousness is being entered by hu- 
manity its appearance or manifestation is first made 
in the most advanced of the race and that only in a 
dim, vague way. This rudimentary condition persists 
for some time, perhaps many thousands of years, then 
the faculty becomes more general in appearance, the 
number of advanced individuals increases, and conse- 
quently, as in the case of the intuitive faculty, it be- 
comes universally prevalent in all humanity; becomes 
transmissible as so-called "acquired characters," and 
then appears as the normal faculty of the entire 
human family cropping out in each individual. Thus, 
in passing from the few advanced ones in the begin- 
ning to that stage where it becomes the common 
possession of all, a faculty requires many thousands 
of years for its perfection, and especially has this been 
true in the past history of the development of human 
faculties. But it is believed that the sweep of the 
life current as it proceeds from form to form, from 
faculty to faculty, gains in momentum as it proceeds, 
so that in these latter years due to the already highly 
developed vehicular mechanisms at its disposal not 
so great a period of time as formerly is required for 
the out-bringing of a new faculty. It might well be 
that while in the past hundreds of thousands of years 
were necessary in the perfection of organs and facul- 
ties, in these latter days only a few thousand, perhaps 
hundreds, may be necessary and that in the days of 
the future not even so many years may be required 
to universalize a faculty. And especially does this 
appear to be true in a state of affairs where so large 
a number of persons are beginning consciously to take 
their evolution in hand and by volitional activities are 



252 THE MYSTERY OF SPACE 

supplying greatly increased impetus to their psychic 
processes which under ordinary, natural methods 
would be considerably slower in their deyelopment. 
It is quite obyious that all cultural efforts when applied 
to the betterment of a given plant, animal or faculty 
result in a corresponding hastening of the process of 
growth far in excess of what that growth would be 
under normal, natural conditions. All the present 
faculties possessed by man are remarkably susceptible 
to cultural influences; in fact, the standing edict of 
ethical and social law is that the human faculties must 
be cultivated as highly as possible, thereby giving the 
spirit a more perfect medium of expression. These 
observations, therefore, lead irresistibly and unavoid- 
ably to the conclusion that the time for the upspringing 
of the intuitional faculty in the human organism is 
even now upon us, that undoubtedly in certain very 
advanced ones it has already reached a notable degree 
of perfection and is rather more general than would 
appear in the absence of careful investigation. 

Now, just as the intellect has made for individu- 
ality, has emphasized the separateness of the Thinker's 
existence from that of other thinkers, has developed 
self-consciousness to a very high degree, even pushing 
it far over into the domain of the higher consciousness 
to the temporary obscuration of the latter, so the in- 
tuitional will make for union, for the brotherhood 
of man, for co-operation and for the common weal. 
Through it man will come gradually into the con- 
sciousness that fundamentally, in his inner nature, in 
every respect of vital concern, he is at-one with his 
fellowmen and not only with the apparent units of 
life but with all life as expressed in whatsoever form 



THE MYSTERY OF SPACE 253 

throughout the universe. Then, too, he will be closer 
to the reality of things, of actions and natural proc- 
esses; in fine, he will have begun the development of 
the space-mind which will bring him to the knowledge 
that he is one with space also and, therefore, with the 
divine life of the world. 

One of the peculiarities of the vital force which 
shows itself in the consciousness as man's intellect, 
is its growing automatism, or that tendency which 
enables the consciousness to perform its functions 
automatically and thus allow opportunity for the de- 
velopment of newer and higher faculties. Actions, 
oft repeated, tend to become automatic. This is also 
true of thought and consciousness. It is one of the 
beneficent results of abstract thought that it develops, 
or tends to develop, a kind of automatism whereby 
a marked saving in time and energy is effected. This 
affords opportunity for other things. It is undoubt- 
edly true that in the days of the truly primitive man 
his consciousness was more completely engaged in the 
execution of the ergonic functions of cells, organs and 
tissues; that all those processes which are now said 
to be involuntary and reflexive were at one time, in 
the distant past of man's evolution, the results of 
conscious volitions. This is a condition which must 
have preceded even the development of the intellect 
itself. Indeed, there could be no intellect in a state 
where the entire modicum of consciousness was being 
utilized in the performance of cellular and histologic 
functions. 

The rise of the intellect must have been in direct 
ratio to the development of automatism among the 
cells, tissues and organs, so that as these came 



254 THE MYSTERY OF SPACE 

gradually to perform their special labors reflex- 
ively the intellect began to be formulated and to 
grow, at first only incipiently, then more and more 
completely until it reached its present state. At the 
present stage of its evolution, a great deal of the 
labor of the intellect is beginning to fall into a kind 
of increscent automatism, although only rudimentarily, 
in many instances. Yet, as a result of this tendency, 
quite the whole of the phenomena of perception is 
characterized by a sort of automatic action. And the 
mind perceives without conscious volition. Many of 
the steps of conceptualization are automatic, in part, 
if not wholly. Certain it is that impulses once set 
in operation whether consciously or unconsciously con- 
tinue to act along the same line until exhausted or 
until the end has been attained. Consequently, it is 
a proven fact that often serious mathematical and 
philosophic problems have been solved by the mind 
long after any conscious effort to solve them had 
ceased. Often solutions have been arrived at during 
sleep. Many such cases might be cited, but the 
phenomenon is now so common that almost every one 
can cite some experience in his own life that will sub- 
stantiate the claim. 

There is no doubt but that these phenomena 
are evidences of a reflexive development in the 
intellect. The time will come undoubtedly, and neces- 
sarily so if the intellect is to give way to a higher 
faculty, which shall be as much above the intellect in 
its grasp of things as the intellect is now above the 
simple consciousness of the lower animal, when quite 
the entirety of our intellectual processes will become 
automatic or self-performing. What then remains of 



THE MYSTERY OF SPACE 255 

the egoic schematism, after its transmutation or ele- 
vation as the organ of the intuitional consciousness 
will be utilized as the organ of the Thinker's invol- 
untary cognitive processes. This will mean that all 
of that laborious ideation which is now the abstract 
thought of the Thinker will be performed auto- 
matically, leaving the higher aspect of the egoic con- 
sciousness free to conceptualize or intuitograph the 
intuitions. Perceptualization then will be replaced by 
conceptualization. This latter will occupy about the 
same status as the former does now. And neces- 
sarily, perception will become more complex. In other 
words, while we now perceive simple percepts which 
are again arranged into concepts making a composite 
picture of the object, we shall then be taking in the 
composite picture of the object at first hand, thereby 
dispensing with the rather slow process of perception 
as it now operates. We shall still be perceiving, but 
what we perceive will be concepts rather than per- 
cepts, as at present. 

The increased powers of intellection gained as 
a result of the increscent automatism in the in- 
tellect, the flowering forth of the intuitive faculty 
and the general enhancement of the intellect through- 
out all its processes will enable it to entertain con- 
cepts or composite picture of things just as readily 
and as perfectly as it can at present deal with a single 
percept. Concepts will be replaced by super-concepts 
or intuitographs. Increased perspicacity will enable 
the Thinker to manipulate the concepts and intuito- 
graphs with the same ease and readiness and withal 
the mind will have attained unto an almost unrealiz- 
able freedom in its search after truth. 



256 THE MYSTERY OF SPACE 

The outcome of this new adjustment which, of 
course, will not spring up at once, but by insensible 
degrees, will be the clarification and unification of our 
knowledge. It will mean also the simplification of it; 
the obviation of diversities of opinions, the springing 
up of a new and winnowed system of philosophy 
which shall be the true one; further, it will imply the 
lessening of the probability of error in our judgments 
and conclusions; the removal of illusion to a much 
larger degree than to-day is possible and the realiza- 
tion by every one of something of the essence of things, 
of causes and effects, of actions, operations, natural 
forces and laws; in fact, a condition of mind which 
will present to the consciousness the simple truth above 
every conceivable phase of kosmic life which may come 
within the scope of the Thinker's observation. 

The further implications of this view are that there 
is a difference between the Thinker and the intellec- 
tuality. The Thinker is eternal and partakes, there- 
fore, of the very essence of primordial originality 
while the mentality is an artificial process, the re- 
sultant of the adaptation of the Thinker's consciousness 
to his vehicular contrivances of objective cognition and 
the interplay of his life among them. 

If the appearance of a choppy sea disturbed by the 
passage of a brisk breeze over its surface be imagined, 
a similitude of the great ocean of life may be en- 
visaged. The wavelet crests symbolize the egos; the 
base of the wavelet which is one with the great sea 
of water represents the Thinker which is one with 
the divine life and consciousness of the kosmos. Just 
as wavelet crests are continually springing up and 
falling back into the sea, so are egos continually being 



THE MYSTERY OF SPACE 257 

cast forth and reabsorbed into the universality of 
life only to be recast, as a wavelet crest or ego, 
upon the surface of the moving ocean of life. And 
so, in this respect, the universum of life and conscious- 
ness which are essentially one is in a constant state of 
ever-becoming, un-becoming and re-becoming. 

Another implication is that, on account of the di- 
versity and complexity of the means of contact with 
the external world, it is not possible for the ego to 
arrive at more than a fragmentary understanding of 
even the latent geometrism of life, mind and materi- 
ality. In our examination of the sensuous world, we 
are very much like the three blind men set to examin- 
ing an elephant. One set to scrutinizing his trunk 
by means of his sense of feeling. When asked for 
his judgment as to what the elephant was he declared 
it was a snake; a second who began with the legs 
found it to be like huge pillars ; and a third who caught 
hold of the elephant's tail and declared the elephant 
to be like a rope. Each one of the blind men described 
what he was able to perceive. To each what he felt 
was all there was upon which he could render judg- 
ment. And so, artists, philosophers, mathematicians, 
musicians, mechanicians, religious seers, metaphysi- 
cians and all other types of mind, are just so many 
blind men set to the examination of an elephant, or 
the sensuous world. Each one confidently believes his 
view to be correct; each one is satisfied with the de- 
liveries of his senses. Yet no one of them is wholly 
correct, no one of them has seen every phase and 
aspect of the problem. Does it not, therefore, appear 
to be the more reasonable and urgent that the view 
which synthesizes the judgments of all the possible 



258 THE MYSTERY OF SPACE 

examiners thereby constructing a composite idea of 
the entire mass of judgments is the more reliable and 
the more correct? 

Referring again to the dual intelligence, the ego 
and the Thinker, which together constitute man, it is 
deemed necessary, in order to present the concept of 
this duality to the mind of the reader in the way that 
shall enable him easily to recall it, to designate the 
egoic intelligence as the egopysche, and the Thinker's 
intelligence as the omnipsyche. 

The egopsyche is the I-making faculty, the faculty 
of self-consciousness and the synthesis of all those 
psychic states and functions known as the intellect or 
mind and includes the ethical aspect of man's nature. 
The omnipsyche is the organism of kosmic conscious- 
ness, the space-mind, or man's higher self and that 
which connects with or allies him to all life; it is the 
basis of human unity and of unity with divinity, just 
as the egopsyche is the basis of separation and in- 
dividuality; it is the organ of direct and instantaneous 
cognition and the permanent essence which has per- 
sisted through every form which the being, man, has 
ever assumed and through every stage of human evolu- 
tion. In it are stored up the memories of the Thinker's 
past, the secrets of life, mind, being, reality, and the 
history of life from the beginning; in it also the plan 
of action for the future of the life-wave as it passes 
from plane to plane, from stage to stage, and from 
form to form. It is the spark from the flame that 
is never quite free from its source; it is the continuous 
spark, the prolonged ray which does not go out and 
cannot be extinguished. It is that in man which when 



THE MYSTERY OF SPACE 259 

full union therewith has been attained makes him a 
god in full consciousness. 

The omnipsyche is really a neglected and over- 
looked factor in the doctrine of evolution. Evolu- 
tionists, while they claim life to be continuous and 
that man has come through all the kingdoms of nature 
in succession and has spent millions of years in the 
perfection of his various organs, faculties and stages 
of consciousness, make no ample allowance for what 
is in reality the basal element in evolution — a continu- 
ous, persisting, permanent life-force which does not 
lose its identity from the beginning to the end of the 
process. This fact — that that spark of life which set 
out upon the evolutionary journey as a moneron has 
glowed steadily from that stage to manhood, main- 
taining meantime its original purposiveness and intent 
— seems to be the most obvious consideration of the 
whole doctrine, yet it has been more or less com- 
pletely ignored. The elementary requirements of 
evolution would seem to establish clearly the necessity 
for some such eternally persisting principle as the 
omnipsyche which is capable of such subtle adaptations 
to every conceivable form of life and in which should 
be gathered up the evolutionary results of every life- 
cycle. For this purpose the omnipsyche or unifying 
principle in man was designed from the beginning and 
it is that which constitutes the basis of his intellectual 
nature while in a far larger sense it is the divinity 
in man himself. It is indeed strange that so important 
a factor as the omnipsyche should have been omitted 
by evolutionists. Yet it can be accounted for upon 
the grounds of the purely mechanistic character of all 
intellectual attempts at solving the problems of vital 



260 THE MYSTERY OF SPACE 

manifestations. But so long as men rely upon 
mechanical explanations of such phenomena so long 
will they be prone to overlook the very essentialities 
of the problems which they devoutly wish to solve. 
The continuity of the physical germ-plasm of the 
human species, 1 now quite generally admitted, would 
suggest, it seems, an analogous condition to the con- 
tinuity of the psychic plasm called the omnipsyche, the 
only difference being that the omnipsyche is an in- 
telligent factor while the physical plasm is a medium 
of transmission though non-intelligent. The omni- 
psyche is, therefore, the psychic reservoir of evolution 
into which are stored the transmuted psychics of 
moneron, amoeba, jellyfish and every other form 
which it has ensouled and acts as the storeroom of 
man's psychic operations as well as the source of his 
intellectuality. 

We turn now from the study of a sketch of the 
mechanism of man's consciousness which gives at its 
best only a fragmentary view of the universe of spa- 
tiality to a consideration of space itself in the light 
of its interrelational bearings upon the question of 
intellectuality. 

In the chapter on the "Genesis and Nature of 
Space" we have, in tracing out the engenderment of 
space, proved it to be basically one with matter (and 
indeed the progenitor of matter), also with life and 
consciousness. Further, it has been shown that all the 
characteristics of materiality are due to the adaptation 
of consciousness to it and that out of this adaptation 
grew the intellectuality. A close approximation to 

*See The Germ Plasm; A Theory of Heredity, by A. 
Weissman. 






THE MYSTERY OF SPACE 261 

this view was maintained by Kant when he discovered 
that our faculty of thinking or the intellect only finds 
again in matter the mathematical order or properties 
which our faculty of perceiving or consciousness has 
deposed there. It appears, therefore, that when the 
intellect approaches matter or spatiality it finds always 
a ready yieldance to its demands simply because in- 
tellectuality has previously established therein the de- 
lineation or map of the path over which it necessarily 
must traverse in its examination of the object of its pur- 
suit. In other words, the kosmic mind in engendering 
materiality and spatiality has set up therein a kosmic 
order or geometrism. Both motor and intellectual 
progress, therefore, can be made through the world of 
spatiality because of the immanence of this kosmic ge- 
ometrism which lies latent in the very fabric of the world 
of substance fashioning both the character and the 
nature of the intellect as well as of space itself. So that 
there is a perfect congruity subsisting between spa- 
tiality and intellectuality. Accordingly it is impossible 
for either one or the other to transcend the grim grasp 
of the mathematical order which binds them in such 
lasting and fundamental agreement. Extra-spatiality 
may degrade itself into spatiality, and indeed in the 
very nature of the case, does so degrade itself, yet 
spatiality can never raise itself beyond the limits set 
by its engendering parent. Materiality may become 
more and more spatialized and consciousness more and 
more intellectualized, but they must proceed hand-in- 
hand one not superseding the other. 

Being the essence of the natural geometry which is 
everywhere immanent in the universum of matter, 
space becomes an organized and ordered extension, 



262 THE MYSTERY OF SPACE 

in fact is the totality of such organized and or- 
dered extension, which conforms to the latent geo- 
metrism the engenderment of which it is the sole cause 
in the last analysis. Does it not appear then that all 
that mass of artificial geometry which has sprung up 
as a result of departures made from this natural 
geometry is utterly baseless and most certainly lacking 
in the kosmic agreement which spatiality lends to our 
primary conceptions? Of course, it is admittedly pos- 
sible to devise certain conventional forms of logic and 
endow them with all the evidences of a rigid con- 
sistency but which, because of their purely artificial 
character, will fall far short of any real conformity to 
the potential geometrism which has been established in 
spatiality. And this fact is of utmost significance for 
all those who seek to find justification either logically 
or naturally for the existence of a multi-dimensional 
quality in space; for, if a clear, discriminative con- 
ception as to the categorical relationship, each to each, 
of the two kinds of geometry be carried in mind, it 
will not be easy to confound them neither will it be 
difficult to discern where the one ends and the other 
begins. 

Now, the fourth dimension and the entirety of 
those mathematical speculations touching upon the 
question of hyperspace, dimensionality, space-curvature 
and the manifoldness of space are purely conventional 
and arbitrary contrivances and do not meet with any 
agreement or authority in the native geometrism which 
we find inhering in space and which the intellect recog- 
nizes there. This conclusion seems to be obvious for 
the reason that, in the first place, the non-Euclidean 
geometries have been constructed upon the basis of a 



THE MYSTERY OF SPACE 263 

negation of the latent geometrism of space and intel- 
lectuality; and if so, is it reasonable to expect that 
either they or any of their conclusions should accord 
with the nature of that form of geometry so admirably 
delineated by Euclid? Obviously not. It is a matter 
of historical knowledge that the whole of the artificial 
non-Euclidean geometries consists of those purely con- 
ventional results which investigators arrived at when 
they denied or controverted the norms supplied by the 
natural geometry. When metageometricians found 
that they could neither prove nor disprove the 
Euclidean parallel-postulate they then set upon the 
examination of idealized constructions which negatived 
the postulate. The results, thus obtained, although 
self-consistent enough, were compiled into systems of 
geometry which naturally were at variance with each 
other and with this inherent geometrism which is found 
in spatiality and answered to by the intellect both 
normally and logically. 

Furthermore, there is another consideration which 
to us seems to be equally if not more forbidding, in 
its objections to the coordination of the two systems 
of geometry, and that is the fact that the geometry 
of hyperspace is denied the corroborative testimony of 
experience and this is true of practically the whole 
of its data. Indeed, there is perhaps no single element 
in its entire constitution which claims the authority of 
experience. This is undoubtedly the weakest point irt 
the structure of the hyperspatial geometries. Conr 
trarily, such is not the case with the natural geometry; 
for, in this, the intellect in retracing its steps over the 
path laid out by that movement which has at the same 
time created both the intellect and spatiality, finds an 



264 THE MYSTERY OF SPACE 

orderly and commodious arrangement into which it 
naturally and easily falls. So exact is this agreement 
of the intellect with the kosmic order that if it were 
possible to remove the whole of spatiality and mate- 
riality there would still be left the frame work which 
is this latent geometrism of kosmogenesis. But the fact 
that the intellect naturally fills all the interstices of ma- 
teriality and spatiality, fitting snugly into all of them as 
if molded for just that purpose, by no means warrants 
the assumption that it would or does also fit the en- 
gendering factor which has created these interstices. 
The frame work, the order or the geometrism of the 
kosmos has been established by life acting consciously 
upon the universum of materiality. And in order to 
establish this geometrism life had to be mobile, active, 
creative. It could not remain static, immobile, and 
accomplish it. Being mobile, dynamic, creative, it 
passes on. It is like a fashioning tool which the cabinet 
makers use in cutting out designs upon a piece of 
wood. It moves, and keeps moving until the design 
is finished, and then it is ready for more designing. 
Life is like that. It cuts out the designs in materiality, 
fashions the form, molds the material, and passes on 
to other forms. The intellect fits into these designs 
gracefully. But what it finds is not life itself, only 
the design which life has made. Hence, as there is 
neither an empirical spatiality nor materiality in con- 
formity with which the artificial geometry of the 
analyst may be said to exist, and as it may not be 
said to conform to the path which life has made in 
passing through either of these, it is absurd to predi- 
cate it upon the same basis as the natural geometry. 
And so, we are forced, in the light of these considera- 



THE MYSTERY OF SPACE 265 

tions to deny the validity and hence the acceptability 
of the non-Euclidean geometries as either reasonable 
or warrantable substitutes for the Euclidean, and deny- 
ing which we also formally ignore the claims of the 
fourth dimension, as mathematically designed, to any 
legitimate anchorage in either our vital or intellectual 
movements. 

It has been shown that the flow of life, as it de- 
scribes that movement which we call evolution, en- 
genders simultaneously and consubstantially spatiality, 
materiality and intellectuality, and these, in turn, the 
natural order or geometrism everywhere immanent in 
the universe; and that automatically, one out of the 
other and each out of the all, these constitute the total- 
ity of kosmic fundamentals. Also we have sketched the 
mechanism of man's consciousness and discovered how, 
in its evolutionary development it has divided into two 
aspects, the egopsychic and the omnipsychic, and these 
two factors ally him definitely and adequately to the 
world of the senses and to the world of supersensuous 
cognitions. And thus we have cleared up some of the 
misconceptions which had to be confronted and made 
more easy the approach to the central idea, thereby 
conserving the substantiating influence which a general 
and more comprehensive view of the whole would 
naturally give. 

The totality of kosmic order is space. It is cir- 
cumscribed by an orderless envelope of chaos just as 
the germ of an egg is surrounded by the egg-plasm. 
The organized kosmos is the germ, kernel or central, 
nucleated mass, enduring in a state of becoming. In- 
volutional kathekos or primordial chaos is the egg- 
plasm which nourishes the germ or the kosmos and 



266 THE MYSTERY OF SPACE 

is that out of which the germ evolves. Kathekos or 
chaos is the unmanifest, unorganized, unconditioned, 
unlimited and undifferentiated plasm. Space is the 
manifest, limited, finite, organized germ that, feeding 
upon the enveloping chaos, exists in a perpetual state 
of alternate manifestation and non-manifestation — 
appearing, disappearing and reappearing indefinitely. 
The appearance of the kosmos as an orderly 
elaboration of the involutionary phase of kosmogene- 
sis, in so far as kosmic order may be said to be an 
accomplished fact, marked the turning point in that 
procedure whose function it was to make manifest a 
universe possessing certain definite characteristics of 
orderliness ; but the kosmos, as it now stands, may not 
be thought of as having attained unto a state of ulti- 
mate orderliness. The idea meant to be conveyed is 
that between the point of becoming and the actually 
pyknosed, or solidified stage in the process of creation 
there is a more or less well defined line of demarkation 
cutting off that which is spatiality from that which is 
non-spatiality. Beyond the limits of spatiality is an 
absence of geometric order. Here geometry breaks 
down, becomes impotent, because it is an intellectual 
construction; at least, it is not so apparent as in the 
manifested kosmos. It is a state about which it is 
utterly futile to predicate anything; because no words 
can describe it. The most that may be said is that 
it is absence of geometric order as it inheres in space. 
And if so, all those movements comprehended under 
the general notions of spatiality, materiality, intellec- 
tuality and geometricity have both their extensive and 
defensive or inverse movements nullified in their ap- 
proach to it. Involutionary Kathekos, therefore, may 



THE MYSTERY OF SPACE 267 

be said to be the primordial wilderness of disorder 
which outskirts the well laid-out and carefully planned 
garden of the spatial universe. We may excogitate 
upon some of the obvious functions of this kathekotic 
world-plasm; but in doing so we must leave off all 
attempts at a description of its appearance, its magni- 
tude, extent or other qualities, and think only of its 
kosmic function. We cannot say that there is back 
of it a spatiality nor can we say that it is a spatiality; 
for whatever may be its extent or volume, it suffices 
that it may not be said to be space. It is chaos. Space 
is order, organization, geometricity. It cannot be said 
that there is a latent geometrism in chaos; because 
geometric order is found only in spatiality and is that 
which distinguishes spatiality from kathekosity or non- 
spatiality. Chaos is the lack of spatiality. This, of 
course, implies that it is impenetrable to the intel- 
lectuality or to vitality. All inverse movement such 
as is discovered as taking place in spatiality and which 
results in the phenomenalization of space runs aground 
when it strikes against the rock-bound coast of kathe- 
kosity. We can only say that it is both the point of 
origin for the evolving universe of life and form and 
its terminus. It is the nebulosity out of which the 
whole came and into which all is ultimately occluded. 
A great and far-reaching error is made in all our 
thinking with respect to the kosmogonic processes 
when we postulate the complete absorption of chaos 
as an early act of kosmogony. Customarily, we think 
of kosmic chaos as a primordial condition whose ex- 
istence was done away as soon as the universe came into 
active manifestation. This because it has been exceed- 
ingly difficult, if not quite impossible, for those whose 



268 THE MYSTERY OF SPACE 

privilege it was to determine the trend of philosophic 
thought to free themselves from the bondage of a 
dogma which owed its existence to a traditional or 
legendary interpretation of facts that ought never 
have been so interpreted. Chaos is and ever shall be, 
so long as the universe itself lacks completion, fullness 
or perfection in purpose, extent and possibility. It 
is undoubtedly being diminished, however, in propor- 
tion as the kosmos is approaching absolute perfection. 
And when the last vestige of chaos disappears from 
the outerskirts of the maturing kosmos there shall ap- 
pear a glorified universe of indescribable qualities. 

Space being a perception a priori cannot be de- 
termined wholly by purely objective methods. The 
yard-stick, the telescope and the light-year are objects 
which belong exclusively to the phenomenal and with 
them alone never can we arrive at a true conception 
of the nature of space. We can no more demonstrate 
the nature of space by the use of objective instruments 
and movements than we can measure the spirit in a 
balance. Certainly, then, it cannot be hoped that by 
taking the measurement of space-distances in light- 
years, or studying the nature of material bodies, we 
shall be able to fathom this most objectively incom- 
prehensible and ineluctable thing which we call space. 
It is such that every Thinker must, in his own inner 
consciousness, come into the realization of that awfully 
mysterious something which is the nature of space both 
as to existence and extent by his own subjective efforts 
unaided, uncharted and alone. When we measure, 
weigh, apportion and otherwise try to determine a 
thing we are dealing with the phenomenal which is no 



THE MYSTERY OF SPACE 269 

more the thing itself than a shadow is the object which 
casts it. 

What does it matter that metageometricians shall 
be able to demonstrate that space exhibits itself 
to the senses in a four- or w-dimensional man- 
ner? Granting that they may be able to do this, 
if merely for the sake of the discussion, when they 
have finished, it will not be space that they have 
determined, but the phenomena of space, its arbor- 
escence, while space itself remain indeterminate and 
unapproachable by phenomenal methods. If there 
are curvature, manifoldness and ^-dimensionality 
these are not properties of space, but of intellec- 
tuality in its cultured state and when it is, there- 
fore, removed from the native state of conception. 
Scientists may be able to weigh the human body, 
count every cell, name and describe every nerve, 
muscle and fiber; they may even be able to know it 
in every conceivable part and from every physical angle 
and relationship, and yet know nothing of the life 
which vitalizes that body and makes it appear the 
phenomenal thing that it is. So it is not by instru- 
ments which man may devise that we shall be able to 
determine the true nature and purpose of space. We 
must adopt other methods and means and assume other 
angles of approach than the purely objective in order 
to comprehend space which, being the sole inherent 
aspect of consciousness, can be understood best by ap- 
plying the measures which the latter provides for its 
understanding. It would appear, therefore, that the 
best study of space is the consciousness itself, knowing 
which we shall know space. 

The universum of space, including the phenomenal 



270 THE MYSTERY OF SPACE 

universe, and its relation to consciousness may be 
likened to a conical funnel whose base represents the 
phenomenal world of the senses and whose apex or 
smallest point represents ultimate reality. 

In Figure 20, we have endeavored to symbolize 
graphically this conception of space. The base marked 
"Sensorium" represents the sensible world. That 
marked "Realism" symbolizes the ultimate plane of 
reality, the inner essence of the world, the plane of 
"things-in-themselves." 

The cone arising from the base "sensorium" sym- 
bolizes the objective world as compared with conscious- 
ness; the subverted cone, with apex in the sensorium, 
represents the evolving human consciousness. 

The successive bases have the following symbology : 
Self-consciousness, Communal Consciousness, Mikro- 
cosmic Consciousness, Makrokosmic or Universal 
Consciousness, the Plane of the Space-Mind Conscious- 
ness, Divine Consciousness, Kathekotic Consciousness, 
or the Plane of Final Union with the Manifest Logos. 

Self-consciousness is that form of consciousness 
which enables the ego to become aware of himself as 
distinguished from other selves or the Not-self; the 
Omnipsychic or Communal Consciousness is that form 
of consciousness from which arises the realization by 
the Thinker of his oneness with all other thinkers and 
with others forms of life. Mikrocosmic consciousness 
denotes a still higher form of consciousness, as that 
which enables the Thinker to become conscious of 
his living identity with the life o£ the world or the 
planet on which he lives. It represents a stage in the 
expansion of consciousness when it becomes one with 
the consciousness of the planet upon which it may 



THE MYSTERY OF SPACE 271 



REALISM 




[Fig. 20. — Kosmos and Consciousnes3 



272 THE MYSTERY OF SPACE 

be functioning. Makrokosmic consciousness accom- 
plishes the awareness of the Thinker's unity with the 
life of the kosmos or universe. The space-mind and 
the consciousness which constitutes it enable the 
Thinker to comprehend the originality and the ter- 
minality of kosmic processes. It is archetypal so far 
as the life-cycle of the universe is concerned because 
the beginning, the intermediate portion and the ending 
of the kosmos are encompassed within it. Divine con- 
sciousness is that form of consciousness which arises 
upon the unification of the Thinker's consciousness with 
that of the manifest deity; it is, in fact, omniscience. 
The kathekotic consciousness belongs to the ultimate 
plane of reality; to kosmic origins and chaogeny, and 
therefore, pertains to the plane of non-manifestation. 
The implications are that in comparison with the 
sensorium, the Thinker's consciousness is a mere point 
in space. It is, in reality, so small and insignificant 
that the extensity of the physical world or universe 
seems unlimited, unfathomable in meaning and infinite in 
extent. But as his consciousness expands, as it passes, 
in evolutionary succession from one plane of reality 
to another and higher one, the illimitability, the in- 
comprehensibility and infinity of the universe grow 
ever smaller and smaller, until the plane of divine 
consciousness is reached. Then the previously incom- 
prehensible dwindles into insignificance, lost in the real 
illimitability, infinity and unfathomability of con- 
sciousness itself. Kosmic psychogenesis, as exhibited 
and specialized for the purposes of the evolution of 
the Thinker, can have no other destiny than the flower- 
ing forth as the ne plus ultra of manifestation which 



THE MYSTERY OF SPACE 273 

is nothing short of unification with the highest form 
of consciousness existent in the kosmos. 

It is not to be considered really that the scope of 
space is diminished but that the growing, expanding 
consciousness of the Thinker will so reduce the rela- 
tive extension of it that illimitability will be swallowed 
up in its extensity. Consciousness, in becoming infinite 
in comprehension, annihilates the imaginary infinity 
of space. Accordingly, that which now appears to be 
beyond mental encompassment undergoes a corre- 
sponding diminution in every respect as the conscious- 
ness expands and becomes more comprehensive. The 
mystery of space decreases as the scope of conscious- 
ness increases. As the Thinker's consciousness expands 
the extensity of the manifested universe decreases. 
Thus the mystery of every aspect of kosmic life les- 
sens, and fades away, as the intimacy of our knowl- 
edge concerning it becomes more and more complete. 
There is no mystery where knowledge is. Mysterious- 
ness is a symbol of ignorance or unconsciousness, and 
that which we do not understand acts as a Flaming 
Sword keeping the way of the Temple of Reality lest 
ignorance break in and despoil the treasures thereof. 

Figure 21 is a graph showing a sectional view of 
consciousness on all planes represented as seven con- 
centric circles. This describes the analogous enveil- 
ment of the consciousness when it ensouls a physical 
body or when bound to the purely objective world of 
the senses. The overcoming of the barriers of reality, 
represented by the circumscribing circles is the work 
of the Thinker who is forever seeking to expand and 
to know. For only at its center, as symbolized here, 
is the consciousness at-one with the highest aspect of 



274 



THE MYSTERY OF SPACE 



kosmic consciousness and there alone is the mystery 
of space despoiled of its habiliments. 

Accordingly, as consciousness or the Thinker is 
more and more divested of carnal barriers and illu- 
sions there develops a gradual recognition of the uni- 
tariness of spatial extent and magnitude; there arises 



Communal 
Consciousness 




Space 

Consciousness 



Fig. 21. — Septenary Enveilment of Consciousness 



the certain knowledge that space has but one dimen- 
sion and that dimension is sheer extension. The 
Thinker's sphere of awareness is represented as if it 
begins as a point in space and develops into a line which 
divides into two lines, the boundaries of the space cones. 
Thus it may be perceived that the ancients had a simi- 
lar conception in mind when they symbolized kosmo- 
genesis with the dot (.), the line, and the circle with 



THE MYSTERY OF SPACE 275 

diameter inscribed, which together represent the uni- 
verse in manifestation. 

We realize the impossibility of adequately depict- 
ing the full significance of the inverse ratio existing 
between the extensity of space and the increscent in- 
clusivity of consciousness by means of graphs; for 
neither words nor diagrams can portray the scope and 
meaning of the conception in its entirety. Yet they 
aid the intellect to grasp a ray of light, an intimation 
of what the Thinker sees and understands interiorly. 

In this connection it is interesting to note the func- 
tion of the ideal in the evolution and expansion of 
consciousness. The ideal has no perceptual value; it 
has no status in the world of the senses. It is unap- 
proachable either in thought or action, and therefore, 
lies beyond the grasp of both the intellectuality and 
the vitality. It is indescribable, inconceptible and 
searchless; for the moment that we describe, define, 
or approach the ideal, either intellectually or vitally, 
in that moment it ceases to be ideal, but actual. It flees 
from even the slightest approach; it never remains 
the same; it cannot be attained, at least its attainment 
causes it to lose its idealty. It is then no longer the 
ideal. It is like an ignis fatnus; the closer we come 
to it the farther away it recedes. It hangs suspended 
before the mind like the luscious grapes which hung 
before the mouth of the hungry Tantalus. As the 
grapes and the water receded from his reach at every 
effort he made to seize them so the ideal remains 
eternally unseizable and unattainable. Whatever, 
therefore, is in our thought processes, or in our knowl- 
edge, that may be said to be ideal, does not really 
exist. The ideal is a phantom growing out of the 



276 THE MYSTERY OF SPACE 

nature and essence of the intellectuality. Its purpose 
is to lead merely the mind on; to allure it, to tantalize, 
and compel it to grow by exertion, by the struggle 
to attain, by the desire to overcome. In this respect, 
it serves well its function in the economy of intel- 
lectual evolution. It is a mysterious aspect of the 
original and eternal desire to live which is the kosmic 
urge present in all organized being and has its roots 
hidden in the divine purpose of creation. 

Idealized constructions, then, are like Arabian 
feasts conjured up by a famishing mentality. They 
are like the dreams of a starving man in which he 
actualizes in phantom-stuff the choicest viands in 
abundant supply for his imaginary delectation. The 
mind that is satisfied never idealizes, never makes an 
idealized construction. It is only when an "aching 
void" is felt, when a longing for the realization of 
that which it has not arises within itself, when a 
feeling of distinct lack, a want, a hankering after 
something not in its reach, takes possession of the 
mind that it begins to idealize. That is why some 
minds are without ideals. It is because they are satis- 
fied with what they have and can understand. They 
feel no hungering for better and grander things; they 
have no desire to understand the unknown and the 
mysterious; hence they do not idealize; they make no 
attempt to represent unto themselves a picture of that 
which is beyond them. Such minds are dormant, 
hibernant, asleep, unfeeling and unresponsive to the 
divine urge. 

But the ideal is neither obtainable objectively nor 
subjectively, neither phenomenally nor really, so that 
when we come upon the ideal in our mode of think- 



THE MYSTERY OF SPACE 277 

ing we have arrived not at a finality or end, but at 
that which is designed to lead us on to something 
higher, to nobler accomplishments and more ex- 
tensive conquests. When we have devised idealized 
constructions, therefore, we should not therewith be 
content but should scrutinize them, examine and 
study them for their implications ; for thereby we may 
discover the path and the guide-posts to a new domain, 
a new ideal, following which we shall, in time, come 
to a point in our search for the real where the 
fluxional is at a minimum; we shall reach that some- 
thing which will admit of no further struggle — the last 
chasm between the phenomenal and the real — and 
standing on the bridge, consciousness, which engages 
the twain, shall have a complete view of that Sacred 
and Imperishable Land of Kosmic Realism where like 
a fleeting cloud of sheerest vapor shall be seen the 
phantom-ideal deliquescing and disappearing in the 
cold, thin air of the real and the eternal. 

Since space is judged to be infinite by the intellect 
occluded in such clouds of illusion and hampered by 
such constrictive bonds of limitations, as it now en- 
dures, we have no right to conclude that the concept 
of infinity would still linger before the mind's eyes 
when the illusionary veil is removed; in fact, there is 
ample reason to believe, nay for the assertion, that 
the recession of the veil will reveal just the opposite 
of this illusion, namely that space is finite, and even 
bounded by the fringe of chaogenetic disorderliness. 
Either we perceive the real or we do not; either the 
pure thingness of all objects can be perceived or it 
cannot be perceived. If not, granting that there is 
such a thing as the real, it must be within the ultimate 



278 THE MYSTERY OF SPACE 

range of conceivability. It also seems reasonable that 
realism exists somewhere, and if so, must be sought 
in a direction inverse to that in which we find the 
phenomenal and the approach thereto must necessarily 
be gradual, continuous and direct and not by abrupt 
breaks, by twists and turns. The phenomenal must lie 
at the terminus of the real, and vice versa. So that 
by retracing the path blazed out by the real in coming 
to phenomenalization we shall perhaps find that which 
casts our shadowy world, just as by tracing a shadow 
in a direction inverse to that in which it extends we 
may find the object which projects it. 

It is not out and beyond that we shall find the end 
of space; it is not by counting tens of thousands of 
light-years that the supposed limits of space shall be 
attained. The path of search must project in an op- 
posite direction — not star-ward but Thinker-ward, to- 
ward the subtle habitation of the consciousness itself. 
We err greatly when we think that by measuring dis- 
tances we shall encompass space; for that which we 
measure and determine is but the clouds caused by 
the vapor of reality. It is, therefore, not without, 
but within, in an inverse direction that the search 
must proceed. Going back over the life-stream, be- 
ginning where it strikes against the shores of solid 
objectivity, deeper and deeper still, past the innermost 
mile-stone of the self-consciousness, back into the very 
heart of the imperturbable interior of being where the 
Thinker's castle opens its doors to the Great Kosmic 
Self, from that open door-way we may step out into 
that great mystery of space — limited, yet not limited, 
multi-dimensioned, and yet having only one dimension, 
veritably real and fundamental, the Father-Mother of 



THE MYSTERY OF SPACE 279 

all phenomena. Here the great mystery of mysteries 
is revealed as the citadel of the universal and the 
ultimate real. In this citadel, the plane of kosmic con- 
sciousness, space loses its spaciousness and time its 
timeliness, diversity its multiplicity and oneness alone 
reigns supreme. 

But the movement towards the center and circum- 
ference of space, after this manner, requires aid neither 
from the notion of space-curvature nor that of the 
space-manifold, except, indeed, only in so far as a 
state of consciousness or a degree of realism may be 
said to be a tridimensional manifold. The feeling 
that space is single-pointed, and yet ubiquitously cen- 
tered, has been indulged by mathematicians and others 
in a more or less modified form; but they have 
imagined it in the terms of an indefinite proceeding 
outward until in some manner unaccountable alike to 
all we come back to the point of origin. It has been 
expressed by Pickering when he says that if we go 
far enough east we shall arrive at the west; far 
enough north we shall come to the south; far enough 
into the zenith we shall come to the nadir. But this 
conception is based upon a notion of space which is 
the exclusive result of mathematical determinations and 
subject to all the restrictions of mathetic rigorous- 
ness. It requires that we shall allow space to be 
curved. This we decline to do for the reason that it 
is both unnecessary and contrary to the most funda- 
mental affirmations of the a priori faculty of the 
Thinkers cognitive apparatus. It would seem to be 
necessary only that we should extend our consciousness 
backward, revert it into the direction whence life came 
to find that which we seek. By extension of conscious- 



28o THE MYSTERY OF SPACE 

ness is meant the ability to function consciously upon 
the various superkosmic spheres or planes just as we 
do on the physical. Yet it should be quite as easy to 
devise an idealized construction which would imagify 
the results of this ingressive movement of the con- 
sciousness as to represent the results of a progressive 
outward movement star-ward. Having done so the 
examination of them could be conducted along lines 
similar to those followed in the scrutiny of objective 
results. 

What would it mean to the Thinker if he were 
able to identify his consciousness with the ether in 
all its varying degrees; what would it mean if he were 
able to identify his consciousness with life and with 
the pure mind-matter of the kosmos; and lastly, with 
the spiritual essence of the universe? What if his 
various vehicles of awareness were available for his 
purposes of cognition? What, indeed, if he could 
traverse consciously the entire gamut of realism and 
consciousness from man to the divine consciousness? 
Does it not appear reasonable that as he assumed each 
of these various vestures of consciousness, in succession, 
he would gradually and finally, come to a full under- 
standing of reality itself? It seems so. This view is 
even more cogent when it is considered that the 
limitations, and consequent obscuration of conscious- 
ness are proportional to the number of vehicles or 
barriers through which the Thinker is required to act 
in contacting the phenomenal universe. Common sense 
suggests that freedom of motility is determined by the 
presence or absence (more particularly the latter) of 
bonds and barriers; that the less the number of such 



THE MYSTERY OF SPACE 281 

barriers the greater the scope of motility and conse- 
quently greater the knowledge. 

Plato evidently had this in mind when he 
imagined the life of men spent in cave-walled prisons 
in which their bodies were so fixed that they were 
compelled to sit in one prescribed position, and there- 
fore, be unable to see anything except the shadows 
of persons or objects as they passed by. He con- 
ceived that men thus conditioned would, in time, suffer 
the diminution of their scope of consciousness to such 
an extent as to reduce it to identification with the 
shadows on the walls. Their consciousness would be 
mere shadow-consciousnesses the entire data of which 
would be shadowgraphs. So that for them the only 
reality would be the shadows which they constantly 
saw. A similar thing really happens to man's con- 
sciousness limited to the plane of the objective world. 
Things which are not objective do not appear as real 
to him, if they do appear at all. It is not that there 
are no other realities than those which appear to the 
egopsychic consciousness or that fall within its scope; 
but that this form of consciousness is incompetent to 
judge of the nature and appearance of those realities 
which do not answer to the limitations under which 
it exists. And so, with men whose data of conscious- 
ness or whose outlook upon the world of facts, or 
rather life, are confined to the narrow bounds of mathe- 
matic rigor and exclusiveness, there may appear to 
exist no realities which may not be defined in the terms 
of mathematics. Similarly, to the empiricist, used 
to measurements of magnitudes, weights, and rates of 
motion, there may also appear to be no realities which 
are not amenable to the mold of his empirical con- 



282 THE MYSTERY OF SPACE 

trivances — the balance, the chromatic and the scalpel. 
All of these are shadow men constricted to the metes 
and bounds of shadows which they observe only be- 
cause they are ignorant of the realities which lie with- 
out their plane. 

Life has so many ways of exhibiting its remains 
to the intellect; and these remains have so many facets 
or viewpoints from which they may be studied, that 
nothing short of a panoramic view of all the modes 
of exhibition and of all the facets and angles of ap- 
pearances will suffice to present a trustworthy and 
comprehensive view of the whole. Then, life itself 
is so illusive, so unseizable by the intellect that the 
testimony of all investigators are required to sum- 
marize its modes of appearance. And, therefore, 
eventual contentment shall be secured only when the 
mass of diverse testimonies is reduced to the lowest 
common divisor, and for this purpose the operations of 
every class of investigators must be viewed as the work 
of specialists upon separate phases, facets and angles 
of life's remains. 

And so it is manifestly absurd for the empiricists, 
by taking note of the dimension, extent, quality and 
character of the shadows, or one single class of angles, 
to hope to predicate any trustworthy judgments about 
either the realities which cast the shadows or underlie 
the angles; because whatever notion or conception 
they may be able to gain must of necessity be merely 
fragmentary and entirely inadequate. Despite this 
fact, however, we still have the spectacle of men who, 
studying the sensible universum of space-content, en- 
deavor either to make it appear as a finality in itself, 
or that the world of the real must necessarily be con- 



THE MYSTERY OF SPACE 283 

formable to the precise standards which they arbi- 
trarily set up in their examination of the objective 
world. It can be said with assurance that we shall 
never be able even so much as to approach a true 
understanding of the unseen, real world until we shall 
have changed our mental attitude towards it and 
ceased to expect that it shall necessarily be fashioned 
and ordered in exactly the same way as the world of 
our senses, or that it shall be understood by applying 
the same methods of procedure as those which we 
use in our examination of the phenomenal, sensuous 
world. It is a matter of logical necessity that, as 
there are no senses which can respond to the real, 
as there are no organs which vibrate in accord with 
the rates of vibration of the real, there can be no 
reasonable hope of understanding it by means of 
sensuous contrivances and standards. 

Let the consciousness, therefore, be turned not 
outward, but inward where is situated the temple of 
divine life ; let there be taken away the outward sheaths 
which enshrine the pure intelligence of the Thinker; 
let him grow and expand his sphere of awareness; let 
there be an exploration of the abysmal deeps of mind, 
of life and consciousness; for buried deeply in man's 
own inner nature is the answer to all queries which 
may vex his impuissant intellectuality. 



CHAPTER IX 

Metageometrical Near-Truths 

Realism is Psychological and Vital — The Impermanence of 
Facts — On the Tendency of the Intellect to Fragmentate — 
The Intellect and Logic — The Passage of Space, the Kos- 
mometer and Zoometer, Instruments for the Measure- 
ment of the Passage of Space and the Flow of Life — The 
Disposal of Life and the Power to Create — Space a Dy- 
namic, Creative Process — Numbers and Kosmogony — The 
Kosmic Significance of the Circle and the Pi-Proportion — 
Mechanical Tendence of the Intellect and its Inaptitude 
for the Understanding of Life — The Criterion of Truth. 

Kosmic truth has many facets. The rays of light 
which we see darting from its surface do not always 
come from the core. Often they are reflections of 
rays whose light stops short at the superfice; and 
these, in turn, are reflections of deeper realities. Thus 
the reflected light may be traced to its source by 
following the lead of external reflections. It is now 
known that moonlight, and perhaps, in many cases, 
starlight, are reflections of sunlight, if not of our sun, 
some other in the universe. But it is only at certain 
times and under certain conditions that we can see 
the sun which is the source of the other kinds of 
light. The stars which owe their light to suns are so 
many facets of sunlight. The moon is a facet of 
sunlight also. Facts are facets of truth. They are 

284 



METAGEOMETRICAL NEAR-TRUTHS 285 

so many faces of eternal truth. They represent the 
many ways reality exhibits itself, or rather its effects, 
to the consciousness. When we, therefore, become 
aware of facts we have not in virtue thereof become 
aware of the reality which produces the facts. We 
have come to know only something of the termini of 
realism while the complexities and internal ramifica- 
tions which lie between realism itself and these termini 
altogether elude our cognition. 

Let us examine briefly an icosahedron, for instance. 
An icosahedron is a figure comprehended under twenty 
equal sides. These various sides are so many faces 
by means of which the figure presents itself to the 
consciousness. These faces, however, are not the real 
object. The figure may be examined by viewing it 
from any one of its sides; yet, by simply examining 
a single face, or any number of faces, less the total 
number, we arrive at no satisfactory knowledge of 
the magnitude or its substance. We must first be- 
come conscious of all the faces, holding them in mind 
as a composite picture, before we can even begin to 
have anything like a complete notion of the icosa- 
hedron. Then by continuing the examination we may 
find that the magnitude is composed of wood fiber 
or stone or metal, as the case may be. In this way 
we might carry the examination to indefinite limits 
and finally arrive at a very comprehensive knowledge 
of the icosahedron and yet be unaware altogether of 
the forces which have been at work in the production 
of the magnitude or of the reality which lies back 
of it. 

Realism is psychological and vital. In essence it 
is mind, spirit, life. Yet these three are one. Mind 



286 THE MYSTERY OF SPACE 

is the outward vehicle of life; spirit is the form or 
the interior vehicle which life assumes in order to ex- 
press itself. Realism, then, is life. Is the logician 
dealing with reality when he collects and coordinates 
the various modes of interpretation by which we learn 
to understand the symbolism of life? Obviously not. 
The data of logic are simply a collection of rules for 
interpreting concepts. It is a compendium of indices 
for the Book of Life. It is no more the book itself 
than a table of contents is a book. But logic occupies 
about the same category as does an index to a volume. 
A book, however, is more than its conventional con- 
tents. It is the thought that is symbolized therein. 
The book of life, accordingly, is the sum total of 
life's expressions; but it is not life itself. That is 
the subtle, evasive something which the contents of 
the book of life symbolize. Nature, both in her pal- 
pable and impalpable aspects, may be said to be the 
book of life wherein are recorded the movements, the 
expressions, and the diacritics of life. The whole is 
a magnitude of many facets (little faces). We shall 
have to know all the faces before we can say that we 
have a comprehensive knowledge of nature. For so 
long as we have only a fragmentary knowledge of 
the whole, so long even as we have merely a super- 
ficial knowledge of any aspect of nature, just so long 
will our knowledge be in vain. Just as it frequently 
happens that, on account of the partial view of things, 
we are led to make incorrect judgments concerning 
them, so when we come to make assertions about life 
or nature in general, we are apt to fall into the error 
of rendering judgments upon insufficient data. And 
it is not at all likely that judgments thus arrived at 



METAGEOMETRICAL NEAR-TRUTHS 287 

can possess true validity because it may happen, and 
undoubtedly does always so happen under the present 
limitations of human knowledge, that the very elements 
which are ignored or neglected in forming a judg- 
ment possess enough of virtue to alter the intrinsic 
value of determinations based upon otherwise insuffi- 
cient data. Hence it develops not infrequently that 
our judgments repeatedly have to be changed in pro- 
portion as our data are made more and more com- 
prehensive. Men searching eagerly for the truth 
sometimes allow themselves to be carried away by the 
enthusiasm of the moment which arises upon the dis- 
covery of a new facet of truth; but if all searchers 
were to bear in mind the fact that reality presents 
itself to consciousness in myriad ways and that there 
are innumerable facets all leading eventually back to 
the source of all they not so easily would be induced 
to jump to the conclusion that they had covered the 
entire ground. For when we have discovered a mil- 
lion facts, or many millions of them, about nature we 
may say that we have only merely begun and that 
what we have found is not to be compared with the 
totality even of the directly observable phases of 
nature. 

Logic, therefore, deals with the symbolism exist- 
ing between and among facets of truth, and not 
directly with truth itself, although the conclusions 
reached by the logicians may be true enough from an 
intrinsic standpoint. Logic is not truth, however; it 
is merely the consistence of relations and inter-rela- 
tions between facts and among groups of facts. Truth 
is not established by logic; it stands in no need of the 
light of logic for its revelation; indeed, more apt than 



288 THE MYSTERY OF SPACE 

not is logic to obscure truth. Truth is its own proof; 
it is self-evident. Logic is a mere modeler of facts; 
it is static, immobile, fixed. All truly logical processes 
need a starting point, a foundation, a premise, a base. 
Truth, being eternal, mobile, dynamic, vital, needs 
no starting point; needs no foundation because it is 
itself fundamental; it requires no premise because no 
premise is comprehensive enough to encompass it. 
There is only one way of arriving at truth and that 
is not to arrive at all — just to recognize it without 
procedure. The fact that facts are, and the fact of 
their relations and inter-relations, their sequence and 
implications, can be arrived at only by logical proc- 
esses. Life, in its passage through the universum of 
spatiality, carefully diacriticizes between the realm of 
facts and the domain of truth, marking each off from 
the other by unmistakable signs and barriers. Truth 
is perceived as an axiomatic, self-evident principle and 
no amount of logic could prove or establish its verity. 
Facts are intellectual creatures; truth is intuitional, 
vital. The intellect conceives the consistence of facts 
while the intuition recognizes truth — is truth, and 
therefore, follows in the wake of life as consciousness. 
There is no permanence in facts and the intel- 
lectual recognition of their consistence. The discovery 
of a single new fact may destroy the consistence of 
a whole mass of previously correlated facts. Thus 
is revealed the miracle power of logic over facts. It 
can take a mass of facts, related or unrelated, mold 
them into hypotheses, endow them with a sort of in- 
terior consistency, and make these hypotheses take the 
posture of truth. Hence logic is often an effective 
mask which the intellect commonly imposes upon its 



METAGEOMETRICAL NEAR-TRUTHS 289 

material; but it does so instinctively and can no more 
escape the rigorous compulsion of this instinctive func- 
tioning than water can escape its liquidity. Where- 
fore, we conclude that true permanence abides alone 
in truth because truth is duration itself. For the 
foundations of the whole structure of facts in religion, 
science, art and philosophy which man has toilfully 
built up in the last million years might easily be de- 
stroyed or overturned by the discovery of some great 
fact or by appreciating the true value of truth. Let 
us suppose it should suddenly be realized by men that 
they are really and truly gods capable of creating and 
possessing all the other virtues, powers and capabilities 
which we are accustomed to impute to supreme di- 
vinity; and suppose that the fact of their omnipotence 
and divine omniscience always had been obvious but 
that men were so engaged upon details and the non- 
essentials of life and matter that they had not noticed 
nor realized it before, would not this realization make 
a vast difference in the character of our knowledge 
and the attitude which we would necessarily assume 
thereafter towards matter, life and the problems which 
they present? Would not it completely revolutionize 
our arts, our sciences and our philosophies? How 
much, then, of the facts of these would be left when 
the light of omniscience had been turned on — when 
truth itself could be perceived and interiorly realized? 
Not much, to be sure. We should undoubtedly have 
to dispense with the entirety of our fact-mass, for it 
should then be entirely useless and meaningless in the 
light of the resplendent omniscience of truth. As at 
present constituted consciousness is focused upon the 
material plane for the purposes of superficial ob- 



290 THE MYSTERY OF SPACE 

servances. But if the focus of consciousness should 
be changed so as to reveal conditions upon what must 
be a higher and more interior level, the aspect of 
things would be entirely changed and the whole of 
our theory of knowledge would have to be recon- 
stituted. It is conceivable, yea obvious, that the stern 
reality of being is far removed into the Great Interior 
of that which is; and there is a point in the path to 
the interiority of being where there is no illusion, no 
appearance, indeed, nothing but the cold, illuminating 
body of reality itself. It must appear also that along 
the journey interior-ward there are many apparent 
levels or planes, each of which requires a new focus. 
It is unreasonable, then, to suppose that the conclu- 
sions arrived at as a result of purely logical processes, 
confined to the lowest levels of reality, are pertinent 
and valid for the entirety of realism which is neither 
of mathematical nor logical import. For instance, if 
we take the purely axiomatic assertion: x equals x, 
the intellect is at once certain that this is so, and can- 
not be otherwise, and yet a proposition of this kind 
is purely conceptual, conventional and arbitrary, x 
may also equal i, 2, 3, 4, or any other quantity. Then, 
if each x in the above equation be replaced, by 
say, a horse, there immediately arises a difficulty. For 
it is not possible to find two horses which are in all 
respects mutually equal. So that as soon as we pass 
from the conceptual into the actual, whether on the 
side of objective reality or that of absolute reality, the 
validity of the axiom is immediately exposed to 
serious questioning. The truth of the matter is that 
on both sides of the conceptual it is always found that 
there is a variance from the standards set up by the 



METAGEOMETRICAL NEAR-TRUTHS 291 

conceptual, this variance being more marked on the 
side nearest to absolute reality than on the side of 
objectivism. Objectively, the conformity of the sensi- 
ble with the conceptual is of such approximation as 
to lend trustworthy utility to the conceptual in its 
application to the sensuous. Thus by simply elimina- 
ting the vital factors from our equations we are en- 
abled to proceed in a reasonably safe manner with 
our judgments. Really, however, no such approximate 
congruence can be found; for on the side of reality 
we are dealing with an indivisible something — some- 
thing that is eternally and absolutely unitary in its 
constitution while when we transfer the scene of our 
observations to the objective world we discover a con- 
trary situation. Here we are everywhere beset by 
diversities, multiplicities and dissimilarities. This is 
so because the intellect naturally tends toward the ob- 
jective where it finds a most comfortable atmosphere 
for its operations. The conceptual is related to the 
objective as a train of cars is related to the railway. 
That is to say, the constitution of the intellect is such 
that it finds its most facile expression in the objective 
world and is about as comfortable in the domain of 
realism as the same train of cars would be on the 
ocean. v 

The intellectuality is designed to deal with facets 
of truth; it is made to manipulate segments, parts, 
fractions, and cannot chart its way through a con- 
tinuum such as reality. Being constitutionally a con- 
ventionality of the Thinker's own contrivance, and 
arising out of the subtle adaptation of his vehicles 
to the environment afforded by the sensuous world, it 
can only find congruence in that conventionality which 



292 THE MYSTERY OF SPACE 

is the instrumentality of a higher intellectuality ex- 
pressed in a diversity of forms, into which reality 
divides itself for manifestation. The human intellect 
is, therefore, the bridge over which is made the pas- 
sage from the individual consciousness to the All- 
Consciousness; simultaneously, the medium whereby 
the physics of the brain are converted into the psychics 
of unconsciousness. It may be likened to a pair of 
specially constructed tongs which are so formed as 
to fit exactly the objects which a higher intellectuality 
has made. It is without the province of the intellect 
to take note of what intervenes between physics and 
psychics; it is always oblivious of interstices while 
taking cognizance of objects or things. In this re- 
spect, the intellect is much like a steerage passenger 
on board an ocean liner who sees only his port of 
departure and port of arrival, knowing nothing in the 
meantime of what happens during the voyage, nothing 
of what the other passengers on the upper decks may 
experience and taking no part in any of the passing 
show until he lands. So that the passage of the in- 
tellect from fact to fact is an altogether uninteresting 
voyage; it may as well be made unconsciously, and 
to all intents and purposes, is so made. 

Accordingly, the advocates of ^-dimensionality 
find it quite impossible to predicate anything whatso- 
ever of the passage, say, from tridimensionality to 
quartodimensionality. They find themselves at ease in 
tridimensionality and have even contrived to find 
pleasant environs in the four-space having made 
therein such idealized constructions as will afford 
ample hospitality to the intellect. But the questions 
as to how the passage from the three-space to the 



METAGEOMETRICAL NEAR-TRUTHS 293 

four-space is to be made and how the intellect shall 
demean itself during the passage have been completely 
ignored and, therefore, left unanswered. What, then, 
shall be said of an explorer who says he has found 
a new land and yet can give no intimation as to how 
one may proceed to arrive at the new land, what 
changes are to be made en route, nor the slightest 
suggestion as to the direction one should take in set- 
ting out for it? It is not likely that the report of 
such an explorer, in practical life, would be taken 
seriously; and yet, there are those who, relying utterly 
upon similar reports made by certain enthusiastic 
analysts, dare to place credence in their asseverations. 
Not only have they given wide credence to these re- 
ports, but have, indeed, sought to rehabilitate their 
own territory in accordance with the strange descrip- 
tions given by unhappy analytical explorers. Now the 
question of greatest concern, granting for the nonce 
that there is such a domain as hyperspace, is the 
passage. How shall we make the passage? Or, is 
the passage possible? In vain do we interrogate the 
analyst; for he does not know, nor does he confess 
to know. Evidently it is impossible for him to know 
by means of the intellect alone; for the intellect not 
being fitted to take cognizance of the "passage," but 
only the starts and stops, has no aptitude for such 
questions. Hence, what seems to be the most im- 
portant phase of the entire question will have to re- 
main utterly inscrutable until the intellect nourished 
by the intuition shall be aroused from its lethargy and 
brought to a certain high point of illumination where 
it, too, may take note of the passage. 

Space is the path which life makes in its down- 



294 THE MYSTERY OF SPACE 

ward sweep through all the stages of pyknosis or 
kosmic condensation by virtue of which it accomplishes 
the engenderment of materiality as also the path 
marked out by it in its upward swing whereby it ac- 
complishes the spiritualization of matter. It is the 
kosmic order which life establishes by means of its 
outgoings and incomings. When we look out into 
space we perceive that which is a dynamic appearance 
of life itself, and not a pure form. Nothing that is 
a pure form can exist in nature and in as much as 
space is not only indissoluble from nature but par- 
takes of its very essence it cannot be said to be a pure 
form. The intellect, however, prone to follow the 
grooves laid out by pure logic, never fails to seek 
to make everything that it contacts conform to these 
logical necessities. But, if the analyst were to make 
careful discrimination as to the respective categories 
— that into which life falls and that in which the in- 
tellect is forced by its nature to proceed — he not so 
easily would be led into the fault of attempting to 
shape realities upon models which being strictly con- 
ventional were not meant for such uses. But neither 
the logician nor the mathematician can be condemned 
for such generosity if such condemnation were justi- 
fiable. For they everywhere and at all times insist 
upon realizing abstractions and abstr actionizing reali- 
ties, and they do this with an insouciance that is at 
times surprising. Yet it is in this very vagary that 
is discovered the true nature of the intellect. There 
is a sort of dual tendence observed in the method of 
the intellect's operation. A polarity is maintained 
throughout: the abstractive and the concretional. It 
vacillates continually between the abstract and the 



METAGEOMETRICAL NEAR-TRUTHS 295 

concrete and no sooner has it found a concrete than 
it begins to set up an abstract for it; and vice versa — 
as soon as it is has constructed an abstract it immedi- 
ately seeks either its concrete or sets out to hew 
some other concrete into such shape as will fit it. 
And between these two extremes numerous excuses 
are found for exercising this peculiar characteristic, 
and that too, without regard to consequences. It 
would seem that the intellect, in thus functioning, was 
really engaged at a sort of sensuous play out of which 
it derived an intense and not altogether unselfish 
pleasure. 

Of course, it must be granted that diversity has 
its specific and withal necessary uses in that it affords 
a field for the operation of human intellectuality and 
represents the adaptation of the kosmic intellect to 
the human for the purposes of evolution. This adap- 
tation while necessary for the intellectual develop- 
ment is, however, not an end in itself. It is merely 
a means to a higher purpose. In fact, if we regard 
materiality as a deposit of life, carried by it as a kind 
of impedimentum, and consciousness, which is life, as 
being identical with the intellectuality which makes 
these adaptations, there should be no grounds for the 
statement that the one is adaptable to the other at 
all. And as this is really the view which we assume 
it would perhaps be more strict to regard the adapta- 
tion as subsisting between the human intellect and 
materiality both of which having been constructed by 
kosmic intellectuality. Pursuant to the diversity of 
uses to which materiality lends itself there arises in 
the intellect a supreme tendency to segment, to break 
up into separate parts, to multiply and diversify. It 



296 THE MYSTERY OF SPACE 

is not content unless it is at this favorite and natural 
pastime. It delights in taking a whole and dividing 
it into innumerable parts. This it will do again and 
again; because all its muscles, sinews and nerves are 
molded in that mold and can no more cease in their 
tendency to fragmentation than can the muscles of a 
dancing mouse cease in their circular twirling of the 
mouse's body. Yet, in this it is but creating a well- 
nigh endless task for itself — which task must be per- 
formed to the uttermost. But in its performance, that 
is, in the intellect's complete understanding of the 
diversity of parts, in the knowledge of their relations 
and inter-relations and in their synthesis, it may arrive 
at that one ineluctable something which is called unity. 
And so doing, become ultimately free. 

In view of the foregoing, it is not surprising that 
the intellect should have, finally, fallen upon the no- 
tion of ^dimensionality. It has come to that as 
naturally as it has performed its most common task. 
Left alone and unhampered in its movements, it has 
simply followed the lead of the Great Highway 
through the domain of materiality. And now it has 
arrived at a stage where it thinks it has succeeded in 
fractionalizing space. Time has long ago yielded to 
fragmentation, been divided into minute parts and each 
part carefully measured. Space, not having a visible 
indicator like time to denote its passage or parts, 
suffered a long and tedious delay before it could boast 
of a measurer. As the sun-dial measured time in the 
past and became the forerunner of the modern clock 
so w-dimensionality measures space for the mathemati- 
cian. What more practical instrument for this purpose 
may yet be devised is not ours to prophesy; yet it is 






METAGEOMETRICAL NEAR-TRUTHS 297 

not to be despaired of that some one shall find a suit- 
able means for this purpose. Seriously, however, it 
is not without possibility that should some subtle mind 
devise an instrument for marking the passage of space 
as we have for denoting the passage of time a great 
stride forward would be accomplished in the evolu- 
tion of the human intellect. For the general outcome 
of the intellect's attention being turned to the passage 
of space would undoubtedly be to recognize not only 
its dynamism but its becoming-ness, as a process of 
kosmogenesis. Because such an instrument would 
have to be so constructed as to take note of the move- 
ment of life, and for this reason, it would have to 
be extremely sensitive necessarily and keyed to the 
subtleties of vitality and not to materiality. Mathe- 
matics shall have failed utterly in the utilitarian aspects 
of this phase of its latest diversion if it do not justify 
its claims by crowning its work in the field of hyper- 
space with a "Kosmometer," an instrument devised 
for the measurement of the movement of space or a 
"Zoometer," an instrument devised for the measure- 
ment of the passage of life. We should like to en- 
courage inventive minds to turn their attention life- 
ward and space-ward with the end in view of con- 
structing such an instrument. When once we have 
learned accurately to measure life we shall then be 
able to dispose of it — to create. It is not doubted 
that if ever humanity is to arrive at that point in its 
evolution where it can understand life; if ever it is 
to attain unto the supreme mastery both of vitality 
and materiality and to come to the ultimate attain- 
ment of divine consciousness (all of which we con- 
fidently believe to be in store for humanity) it must 



298 THE MYSTERY OF SPACE 

be accomplished after this manner: first, by syncretiz- 
ing materiality with vitality, and then, by intuitionally 
recognizing the truth of the implications of the syn- 
cretism. 

The history of consciousness in the human family 
is identical with the history of man's conquest over 
matter and physical forces. And this is clearly indi- 
cated in the incidentals contingent upon the toilsome 
rise of the genus homo from the earliest caveman 
whose status denoted a comparatively negligible tran- 
scendence of material forces, to the present-day man 
who has gained a markedly notable conquest over 
these forces. Always consciousness seeks the means 
of adequately expressing itself in the sensible world. 
And to this end it engenders faculties, organs and 
processes in the bodily mechanism, and, in matter, 
devises instruments of application whereupon and 
wherewith it may test, analyze, combine and recom- 
bine the forces and materials it finds. The unlim- 
ited range of expressions lying open to the conscious- 
ness makes it necessary continually to devise higher 
and higher grades of appliances to meet its needs as 
it expands. It will not be gainsaid that the telescope 
has served actually to lay bare to the consciousness 
an immeasurable realm of knowledge nor that the 
microscope, turning its attention in an opposite direc- 
tion, marvelously has enlarged and enrichened our 
knowledge of the world about us. And similar decla- 
ration may be made anent almost every invention, 
discovery and conquest which man- has made over 
natural phenomena. Thus, by externally applying 
mechanical implements to the subject of his conscious- 
ness, man has extended actually his consciousness, his 



METAGEOMETRICAL NEAR-TRUTHS 299 

sphere of knowledge ; has greatly enhanced its quality, 
and, in the process, has urged the intellect to en- 
deavors that have wrought its present unequaled 
mastery of things. Nor have the spiritual aspects of 
our advance along these lines been the least notable. 
For these have enjoyed the essence of all that has 
been gained in the process and have, therefore, kept 
pace with the onward movement of the intellectual 
consciousness. But heretofore no advance has been 
made as a result of methodic or reflexive determina- 
tions. That is, men did not set out from the beginning, 
equipped with foreknowledge of what their efforts 
would bring, to develop the present quality of human 
consciousness. They simply worked on, their atten- 
tion being absorbed by the problems that lay nearest 
and demanded earliest consideration. So the advance 
has come as a resultant of man's close application to 
his ever-present needs — shelter, clothing, food, pro- 
tection and other preservative measures — and it has 
come naturally and inevitably and without prepense. 
Nevertheless, if man, knowing what to expect from 
the syncretization of matter and mind, after this 
fashion, should set out deliberately to accelerate the 
intensification, expansion and growth of his conscious- 
ness, there is no doubt but that the consequence would 
be most far-reaching and satisfactory. 

But the path that leads to this grand consumma- 
tion does not lie in the direction of diversity; it lies 
in the opposite direction. In vain, then, does the 
intellect fractionalize in the hope that by doing so it 
shall come to the solid substructure of life; in vain 
does the analyst segment space into any number of 
parts or orders; in vain does he ask how many and 



3 oo THE MYSTERY OF SPACE 

how much; for by answering none of these queries 
will he find the satisfaction which he vaguely seeks. 

If it be true that it is not by analysis but by syn- 
thesis that the true norm of life, and therefore, of 
reality shall be found it is futile to entertain serious 
hope of finding it in any other way. As a perisophism 
or near-truth, then, w-dimensionality takes foremost 
rank. And this is so for the reason that when we 
proceed in the direction of multiple dimensions, that 
is, one dimension piled upon another dimension or 
inserted between two others we are traveling in a 
direction which, the more we multiply our dimensions, 
leads us farther and farther away from the truth. 
This is a simple truism. If we take, for instance, a 
wooden ball and cut it up into four quarters, and 
divide each one of these quarters into eighths, into 
sixteenths, thirty-seconds, sixty-fourths, etc., indefi- 
nitely, we shall have a multiplicity of parts, each one 
unlike the original ball. But from no examination of 
the multipartite segments can we derive anything like 
an adequate conception of the original ball. Some- 
thing, of course, can be learned, but not enough to 
enable the rendering of a correct judgment as to the 
nature, size, shape and general appearance of the ball. 
But this is precisely what happens when the analyst 
divides space into many dimensions. He cuts it up 
into ^-dimensional parts and the more minutely he 
divides it into parts the more remote will each part 
be in its similarity to the original shape and form of 
space, and the farther away from the true conception 
of the nature of space he is led thereby. 

Now, ^-dimensionality or that phase of meta- 
geometry which regards space as being divisible into any 



METAGEOMETRICAL NEAR-TRUTHS 301 

number of dimensions or systems of coordinates is a 
direct and inevitable product of that tendency of the 
intellect to individuate and to singularize phenomena. 
Biologically speaking, it is a peculiarity which harks 
back to the time when life was manifested through 
the cell-colony and when the individual cells began, 
because of increasing consciousness, to detach them- 
selves from the colony and set out for themselves, and 
thus each intellect recapitulates in its modus vivendi 
the salient tendencies of phylogenesis. Let it suffice, 
then, to point out that this universal tendency to seg- 
ment and fragmentate which rigorously characterizes 
intellectual operations upon every phenomenon with 
which it deals is a culmination of the primordial 
tendency among cells to divide, inasmuch as this phase 
of cell life must be the work of the kosmic intellect. 
The natural inference is that from the extreme of in- 
dividualization there shall be a gradual turning, 
whether of the intellect per se or of the intellect joined 
to the intuition does not matter, towards that other 
extreme of communalization. And from this latter 
shall grow up, as one of the inevitable and ineluctable 
tendencies of the Thinker's consciousness a torrentious 
movement in human society towards cooperation, 
brotherhood, mutuality and union in everything. So 
that whereas in the past and at the present time the 
intellect has been developing under the dominant note 
of individuality it will then be coming gradually under 
another dominant note — communality. The result of 
this development will be the unification of all things, 
and instead of many dimensions of space, many 
measures of time, and a general diversification of all 
phenomena, we shall come to the only true notion of 



302 THE MYSTERY OF SPACE 

these things and realize pragmatically the true value 
and extent of unity in the universe. 

It is admitted that the intellect, in treating objects 
singly and dealing only with the starts and stops of 
a movement, is withal loyal to the kosmic order, design 
and purpose which have priorly characterized mani- 
fested phenomena by segmentation. And in this loy- 
alty it has been following merely a natural lead which, 
while admitting of the widest development and ex- 
perience, nevertheless at the same time beneficently 
obscures the underlying reality in order that in its 
adaptation to the sensuous world the intellect might 
have the greatest freedom for the development suited 
to the given stage of its evolution. But in thus ad- 
mitting the natural congruence between the intellec- 
tuality and the phenomenal or sensuous we do not 
thereby unite with those who already believe that this 
kosmic agreement is the ne plus ultra of psychogenesis. 
On the other hand, it is maintained that this is merely 
a phase of psychogenesis which shall be outgrown in 
just the same measure as other phases have been out- 
grown. And notwithstanding the fact that judgments 
of the intellect with respect to inter-factual relations 
or the ens of facts themselves are as valid as its judi- 
cial determination of self-consciousness, no more and 
no less, we are, by the very rigor and exclusiveness of 
this logical necessity and inherent limitation, led to 
view the intellect's interpretation of phenomena as 
partial and fragmentary; for the reason that the neces- 
sitous confinement of its understanding and interpre- 
tative powers to fact-relations quite effectively inhibits 
the use of these powers for the contemplation of the 
deeper causative agencies which have operated to pro- 



METAGEOMETRICAL NEAR-TRUTHS 303 

duce the phenomena. But it is apparent that just as 
the transmuted results of other phases of psychogenesis 
are now being utilized as a basis for the efficient 
operation of the intellect in the sensuous world, thereby 
enabling the attainment of a very high mastery over 
matter, so will the functional dynamism acquired by 
it in the pursuit and comprehension of diversity serve 
well when, in later days, it has acquired the power 
to deal directly with reality, to create and dispose of 
life just as the kosmic intellect has and is now using 
it in the execution of the infinite process of becoming 
through which creation is proceeding. It would seem 
that the necessary prerequisite to the development of 
any higher functional capability is that the intellect 
should be capable of disposing of innumerable details, 
indeed the totality of kosmic detail, before it can 
come wholly into the power and capacity to under- 
stand and manipulate life. Furthermore, it appears 
that the acquirement of this power quite necessarily 
has been delayed awaiting that time when, dominated 
by the intuition, the intellect shall have attained the 
requisite managerial ability for marshaling an ex- 
ceedingly large number of details. 

The supreme tendency of life is expression. And this 
expression, singularly enough, reaches its most perfect 
phenomenalization by means of that movement which 
results in the multiplication of forms. Despite the fact, 
therefore, that the comprehension of reality involves 
a gradual turning away from the exclusive occupation 
of organizing a multitude of separate and apparently 
unrelated facts to a monistic view which at once 
recognizes the unitariness and co-originality of all 
things, of life, mind and form, the intellect will need 



3 o 4 THE MYSTERY OF SPACE 

the training and development which come from vhe 
mastery of diversity. It is, then, not difficult to per- 
ceive the wise utilitarianism of the present schematism 
of things as shown in the universal tendency in the 
intellect to devote itself exclusively to parts or segments 
of truth. 

Whenever an individual intellectuality, on account 
of prolonged thought and the consequent inurement 
of the mind to higher and higher vibrations of the 
kosmic intellect, brings itself to such a high point of 
sensitiveness that it can receive so much as an intima- 
tion of some great truth, it begins to sense, in a more 
or less vague way, something of the substance and 
general tendence of the underlying reality of that 
which foreshadows its appearance. Then, confounded 
by the multiformal characteristics of kosmic truth be- 
cause of the fact that it presents itself in such numer- 
ous ways and forms, men often are induced to attempt 
the reformation of all facts, or a great mass of 
kindred facts, in accordance with the newly-found fact 
or principle. They forget evidently that no fact in 
the universe can be at variance with any other fact 
and still be a fact. So that in the totality of facts 
every separate and distinct fact must be congruent 
with every other fact forming a beautiful, harmonious 
and symmetrical whole; but often the whole is made 
to suffer in the attempt at making it conform to the 
substance of a mere intimation. Moreover, it is con- 
ceivable that even the totality of facts may lack a 
rigid conformity with reality in all its parts and that 
having compassed the entire mass of facts one may 
fall short of the understanding of realism. 

This is practically what has happened in the mind 



METAGEOMETRICAL NEAR-TRUTHS 305 

of the metageometrician who having received an in- 
timation as to the real nature of space as that whose 
center is everywhere and yet nowhere and whose 
nature is psychological and vital rather than mathe- 
matical and logical, misses the great outstanding facts 
and clings to the intimations which he experiences as 
to the nature of space. He, therefore, concludes that 
the form of space is that of a flexure or curve. 
There is a valid element in the notion of the curvature 
of space but not enough of truth wholly to validate 
the notion. Since the very reality of space is a matter 
which can be determined only by the conformance of 
the consciousness with it in such a manner as to render 
the conception of it entirely unintelligible to the in- 
tellect except in so far as it may be able to identify 
itself with the space-process, there is much room for 
the serious questioning of the mathematic conclusion 
upon the grounds of its fragmentariness if not entirely 
upon the basis of its invalidity. Wherefore it may 
be seen that any search for either the center or the 
extreme outer limits which proceeds in a manner con- 
formable to the external indications of the intellectual 
order is vain, indeed. Although it is undoubtedly true 
that the attainment of a central or frontier position 
in space does not involve any lineal progression what- 
soever, the same being attainable, not by progression 
nor by overcoming distances, but by a subtle adjust- 
ment, yea, a sort of attachment of the consciousness 
to the order of becoming which binds the appearance 
of space, wherever one may be, it is nevertheless diffi- 
cult and painful for the intellect to grasp the totality 
of this truth at one sweep. Indeed, it is not possible 
for it, alone and unaided by the intuition, to grasp 



3 o6 THE MYSTERY OF SPACE 

it at all. Hence, the mathematician who depends en- 
tirely upon the deliveries of the intellect which con- 
form, in their passage from the conceptual to the 
written or spoken word, to all the rigors of mathetic 
requirements, fails utterly in perceiving the magnitude 
of this conception and all its connotations; he fails 
because his prejudices and the woof and warp of his 
intellectual habits prevent his assuming a sympathetic 
attitude toward it and thereby precluding at the start 
any calm consideration of it. And not only is this 
true of the mathematician but of all those whose en- 
deavors are confined to the plane of purely sensuous 
and logical data. It would, therefore, appear that 
our entire attitude towards things spatial must be 
changed before we can even begin to perceive the 
reality which is really the object of all researches in 
this domain. But, on the surface, there is after all 
little difference between the ultimate facts involved 
in these two totally different conceptions. Mathe- 
matically speaking, all progression eastward would 
terminate at the west, and vice versa; and the same 
would be true regardless of the point from which pro- 
gression might originate. Always the terminus would 
be the opposite of the starting point. Then, too, it 
might be said that if we sought the space-center we 
should arrive at the circumference. The difficulty with 
this view is that there is a very remote, though im- 
portant, connection between it and the truth of the 
matter. But the partiality of this view, and the ab- 
sence of either experience or intuition to intimate a 
more reasonable view, serve effectively to buttress it 
as a hypothesis acceptable to many. Thus it is ever 
more difficult to supplant a near-truth than it is to 



METAGEOMETRICAL NEAR-TRUTHS 307 

gain credence for the whole truth. On the other hand, 
according to the view which we maintain here, it is 
quite true that the seeking of the kosmic space-center 
will reveal the circumference; that the search for the 
nadir will uncover the zenith; the east effloresces as 
the west, and a northward journey will wind up at 
the south, etc., but in quite a different manner from 
that which the mathematician has in mind when he 
postulates the curvature of space. Our view involves 
no space curvature nor any other spatial distortion. 
// deals with space as reality, as a dynamic process, 
a flux which, like the sea, is continually casting itself 
upon the shores of chaos and falling hack upon itself 
only to he recast against the rock-hound coast of its 
chaotic limits. Now, that which falls back upon itself 
and rolls in a recurrent movement upon its own sur- 
face is life which, in its recession is the natural and 
kosmic limitations of itself, generates matter in all 
its varied expressions. Space, m its extensity, cannot 
transcend life; for it is the path which life makes in 
its out-coming, its manifestation. Of the chaotic fringe 
which circumscribes the manifested universe it is ab- 
surd to say that it is vital or psychological in any 
sense of these terms. For notwithstanding the fact 
that out of its very substance are engendered life, 
intellectuality, spatiality and materiality, it is never- 
theless none of these in its primary essence. It is 
Chaos-Kosmos; because from its content the kosmos 
is evolved, and it still remains; it is chaos-spatiality; 
chaos-materiality ; chaos-intellectuality ; chaos-geometric- 
ity; because these are engendered by the movement 
of life in chaos while at the same time there remains 
a residuum of the chaogenetic substance which consti- 



3 o8 THE MYSTERY OF SPACE 

tutes the limitations of all these subsequent processes. 
In this sense, the chaogenetic fringe becomes the limits 
of the manifested universe so that it would appear 
that all those major processes outlined above are finite 
manifestations of the eternal chaos. But none of those 
possibilities of motion which are found in these major 
movements of the kosmos can be logically said to 
exist in chaos. It is the embodiment of everything 
that is the opposite of those qualities which may be 
found in them, that is, in materiality, vitality, spa- 
tiality, intellectuality and geometricity. 

Apropos to this phase of the discussion let us 
examine briefly one of its most significant implications, 
both mathematical and kosmic, which arises out of the 
fact that space is an engendered product of life that 
is bound by the fringe of chaos which sustains and 
limits it. The chaotic fringe plus manifesting kosmos 
constitute 5 the absolute magnitude of the kosmos. The 
manifestation factor is complemented by the chaos 
factor and together the two define the full universe. 
Kosmogony is the universal movement of all kosmic 
elements or factors in diminishing the chaotic com- 
plement and reducing it to kosmic order or geometrism. 
It is undoubtedly impossible to determine mathe- 
matically the exact volume of either complement or 
the ratio of the one to the other; yet it is conceivable 
that the chaotic fringe is greater in extent than the 
ordered portion of the kosmic uni-circle or universe. 
It is even conceivable that the difference, upon the 
basis of the meaning of the Pythagorean Tetragram- 
maton and the view outlined in the Chapter on the 
"Mystery of Space," is as seven to three wherefrom 
the conclusion might be drawn that the universe has 



METAGEOMETRICAL NEAR-TRUTHS 309 

yet seven complete stages more or less of evolution 
before the close of the Great Cycle of Manifestation 
when the fringe of chaos shall have been totally used 
up in the work of creation. But for those who may 
experience impatience at the infinitude of the process 
when viewed in this light the terms may be reversed 
and the difference may be conceived as the ratio of 
three to seven wherefrom the conclusion would fol- 
low that the kosmogonic process is seven-tenths com- 
plete, as it will not vary the seeming infinitude either 
way it may be determined. The notion, despite its 
speculative character, offers an explanation of other- 
wise inexplicable conditions, and, on account of its 
profound connotations, may even be found to be pro- 
ductive of the highest good in its equilibrating in- 
fluence upon our mode of thinking. 

In any event, there does appear to be a subtle 
relation subsisting between true numbers and kos- 
mogony. Number is a phase of the kosmogonic move- 
ment, a measurer of the intellect and the establisher 
of the geometrism of space, answering tentatively to 
the numericity of pure being. In fact, being actually 
expresses number and number itself is an evolution 
and not a thing posited once for all as a pure, in- 
variable form in the universe. It is, like the kosmos, 
in a state of becoming and there may yet appear to 
our cognitive powers a whole series of new numbers 
pure in itself and altogether conformable to the con- 
ditions reigning at the time. 

The symbology of the circle, in all times recog- 
nized as the true symbol of the kosmos in eternity, 
of eternity itself, of the archetypal, of space, duration 
and Ultimate Perfection, is replete with profound sig- 



310 THE MYSTERY OF SPACE 

nifications. But it should be understood that the 
circle is a symbol of the perfected universe and not 
the universe in a state of evolution. It symbolizes 
perfection, completion and the ultimate union of the 
manifesting with the archetypal which results in the 
crowning deed of Perfection. The circle is, therefore, 
not a symbol of the universe as it now stands ; it does 
not represent a snapshot view of the kosmos but the 
universe as a full. It cannot be a full until it has 
attained the ne plus ultra of completion; for a kosmic 
full is that state to be attained by the manifested 
kosmos upon the termination of all the fundamental 
processes now in operation. But it is this state that 
the circle really represents, and by virtue of which it 
possesses its intrinsic qualities and also in virtue of 
which the intellect recognizes these qualities. The 
properties of understanding and recognizance in the 
intellect are veritably fixed by the status quo of the 
universe during every stage. That is, the focus of the 
intellect, like the focus of a chromatic lens, is adjusted 
by the fiat of the nature and eternal fitness of things 
to correspond exactly with every state through which 
the kosmos itself passes. This is one of the obvious 
implications of the phanerobiogenic behavior of the 
kosmos and is necessarily resident in the notion of 
the genesis of space and intellectuality as consubstan- 
tial and coordinate factors. 

Wherefore the more cogent is the reason for the 
belief that the inherent qualities of the kosmogonic 
fundamentals; as, vitality, materiality, spatiality, in- 
tellectuality and geometricity, are true variants, and 
that their variability is proportional to the progress 
of these major movements toward the ultimate satis- 



METAGEOMETRICAL NEAR-TRUTHS 311 

faction of the original creative impulse. May it not 
be, therefore, that the indeterminate character of 
the ratio of the diameter to the circumference 
(3. 141 5926 . . .), 1S due t0 causes far more pro- 
found than the crudity of our micrometers or the 
mere supposed fact of the circle's peculiarity? May it 
not also be true that the pi proportion shall become 
a whole number, and in its integration, keep apace 
with the perfecting process of the kosmos, diminishing, 
by retrogression to one or increasing, by progression, 
to ten which, after all, is essentially unity, being the 
perfect numeral? It is not without the utmost as- 
surance that these queries will be categorically ques- 
tioned by the orthodox, creed-loyal, strictly intellec- 
tual type that we sketch these implications, but it is 
felt to be an urgent duty to remind all such that the 
most effective barrier to realization in the field of 
philosophy is an intolerant attitude towards all lines 
of thought which suggest the impermanence of con- 
ditions as we find them in the kosmos at the present 
time. The fact is that our lives are so distressingly 
short that we have neither time nor opportunity to 
watch the changing moods of the kosmos nor discern 
the gradual reduction of mere appearance to the firm 
basis of reality, and accordingly, the intellect tena- 
ciously clings to those notions which it derives from 
the instant-exposrre which the lens of intellectual con- 
ceivability allots to it. Once the view is taken it is 
immediately invested with everlastingness. This 
everlastingness is then imputed to the kosmos in that 
particular pose, attitude or state. Always the intel- 
lect beholds in that passing view, snatched from the 
fleeting panorama of eternal duration, a picture of 



312 THE MYSTERY OF SPACE 

itself which it mistakes for the reality of the not-self. 

The inclination of the axis of the earth toward 
the plane of its orbit is approximately twenty-three 
and one-half degrees. No well-informed astronomer, 
however, doubts now the fact that this ecliptic angle 
is being gradually lessened; because, as a result of 
centuries of observation, it has been found to be de- 
creasing at the rate of about 46.3 seconds per century. 
Yet no intellect is able to perceive in any given life- 
time the actual decrement of this angle. It is only 
by careful measurements after centuries of waiting 
that a difference can be discovered at all. Thus it 
may even be so with the ratio of the diameter to 
the circumference of a circle, the only difference being 
that it has not yet been determined whether there is 
a decrement or an increase in the size of the ratio. 

The pi proportion is, then, a register or measurer 
of the slow, measured approach of the manifesting 
kosmos to the standard of ultimate perfection. There- 
fore, and in view of these considerations, we may 
not hesitate to confirm our belief in the validity of 
the notion that it actually and literally expresses the 
key to the evolutionary status of kosmogony. The 
mathematical determination which limits it as an un- 
changeable, inelastic quantity is, consequently, only 
partially true and leads to the inclusion of this quan- 
tity under the category of mathematical near-truths, 
for such it appears to be in spite of its rigorous estab- 
lishment. 

The formal topography into which the intellect 
spreads when seeking the ideal and the abstract is 
not a condition which is derivable from the real es- 
sence of life or matter, but, on the other hand, is a 



METAGEOMETRICAL NEAR-TRUTHS 313 

product of the intellect itself partaking of its nature 
rather than of the nature of reality. There is, there- 
fore, a very important distinction to be made between 
all deliveries of the intellect and the realism both of 
the objects and conditions to which the intellectual 
deliveries pertain. One of the most marked pecu- 
liarities of the human intellect is the fact that it 
always unavoidably stamps its own nature and features 
upon every datum which passes through it to the con- 
sciousness. The utmost importance attaches to this 
phenomenon, for the reason that it points to the 
necessity of carefully scrutinizing intellectual deliv- 
eries and the making of allowances for those ever- 
present characteristics which the intellect superim- 
poses upon its data. Perhaps the inherent colorific 
quality which it imposes upon our knowledge would 
be better understood if a similitude were indulged at 
this juncture. The intellect may be likened to a 
color-bearing instrument which, when it has once 
handled an object, leaves forever its own color trans- 
fused into every cell and fiber of the object so that 
when the same object is presented to the conscious- 
ness for purposes of cognition it bears always the same 
peculiar marks and colorations which the intellect, in 
its manipulation of it, places thereon. In this respect 
the intellect may also be said to be like a potter who has 
but one mold and that of a peculiar formation. 
Hence, whatever wares it presents to the consciousness 
will invariably be found to be molded in conformity 
with that particular mold. If it were possible to view 
reality or the essential nature of things the difficulty 
which now the intellect lays in the path of direct and 
uncolored cognition would be obviated; for then there 



3H THE MYSTERY OF SPACE 

no longer would be any necessity of viewing things 
as they are colored or molded by the intellect. The 
intuition, being a process of pure consciousness, will, 
when it has arisen to a position where it may dominate 
the intellect as the intellect now dominates it, so 
modify this tendency which we see so ineradicably 
bound up in the very nature of the intellect that the 
apparently insurmountable difficulties which it has in- 
terposed between mere perception and a direct cog- 
nitive operation will be quite completely overcome. 
Thus, in the above, is discovered another obstacle 
which posits itself between the notion of space as 
reality and the intellectual determination of it which 
the mathematician examines and to which his con- 
sciousness is necessarily limited. Furthermore, it may 
be perceived also how easily the mind may be deluded 
into thinking that the intellectual notion which it en- 
tertains of space is necessarily correct, when obversely, 
it is simply examining a concept which has been re- 
made by the intellect into a form which is not at all 
unlike its own peculiar nature, and therefore, as much 
short of reality as the intellect itself is. Similarly, 
if the mathematical mind succeed in catching a glimpse 
of the reality of space in the form of an intimation, 
which, in itself though fragmentary, is nevertheless 
true, its consciousness is finally deprived of the 
true validity thereof simply because of the behavior 
of the intellect in its manipulation of it. The im- 
portance of these intellectual difficulties cannot be 
over-estimated for they furnish the -grounds for the 
ineptitude of intellectual determinations made in a 
sphere of motility to which the intellect is a stranger. 
And this fact will appear more evident when it is 



METAGEOMETRICAL NEAR-TRUTHS 315 

perceived that quite the entire content of human 
knowledge has been thoroughly vitiated by them. So 
that only in those very rare moments which (in a 
highly sensitive mentality) enable the intuition to gain 
a momentary ascendancy over the intellect is it pos- 
sible for the Thinker to catch hold of realism itself, 
and project the truth of what he sees into the lower, 
intellectual consciousness. But so small is that portion 
of our knowledge which owes its origin to the intui- 
tion that when compared with the totality of that 
which we seem to understand it is well-nigh negligible. 
And then, when it is considered that at present there 
is no way of conceptualizing adequately the intuito- 
graph so as to make it propagable the insignificance 
of this form of knowledge is even more notable. It 
can now be seen in how large a measure the notion 
of the curvature of space is merely an intellectual 
translation of a true intuition into the terms of the 
intellect which, in the very nature of the case, can 
only approximate the truth because of its colorific 
habits. 

A similar declaration may be made of that other 
datum of metageometrical knowledge which postulates 
the ultimate convergence of parallel lines. In fact, 
what has been said as to the perisophical nature of 
the notion of space-curvature will apply with equal 
force to the idea of parallel convergence since the 
latter is a derivative of the former. But there is yet 
another consideration, apart from the colorific in- 
fluence of the intellect, which, although it partakes 
of the nature of this quality, is nevertheless a near- 
truth of quite a different order. This may be better 
understood by referring to the graph showing the 



316 THE MYSTERY OF SPACE 

inverse ratio of objective space to the conscious- 
ness. 1 Let us suppose that the graph may also 
represent the Thinker's outlook into the world of 
spatiality. It then appears that, because of that move- 
ment of consciousness in its pursuit of life which, as 
it expands, makes the objective world to appear to 
be diminished in proportion to the extent of its ex- 
pansion, it is quite natural, under such circumstances, 
that parallel lines drawn anywhere in the limits of 
the objective world should seem to come to a point 
in the ultimate extension of themselves. While this 
graph is not meant to depict such a view, it may be 
found nevertheless, to be a true delineation of the 
topography of that state of mind into which the meta- 
geometrician brings himself when he visualizes space 
as curved; for there is no doubt but that a state of 
intellectual ecstasy, such as that in which the mind of 
the metageometrician must be functioning in order to 
perceive space in that form, is quite different from 
the normal and, therefore, in need of a different topo- 
graphical survey. But, if we grant that in the crea- 
tional aspects of space there is conceivable an ever- 
present tendency to convolution, or a rolling back upon 
itself, it is imaginable that parallel lines inscribed 
either upon its surface or in its texture need not neces- 
sarily meet but maintain their parallelism regardless 
of the complexity of the convolutions. The con- 
vergence of parallel lines is much like a tangent in 
the outgrowth of the idea from the notion of space- 
curvature. The more a tangential line is extended the 
farther away from the circumference it becomes and 
consequently less in agreement therewith. The more 

1 Figure 20. 



METAGEOMETRICAL NEAR-TRUTHS 317 

subsidiary propositions or corollaries are multiplied 
the more remote from the truth the determinations 
become and especially is this true of the hypothesis 
of space curvature. 

In the notion of the manifoldness of space, by 
virtue of which it is conceived as existing in a series 
of superimposable and generable manifolds of varying 
degrees of complexity, are discernible traces of that 
intuitional intimation which underlies the assertion 
that because of the necessary phenomenalization of 
reality for the purpose of manifestation to the intellect 
it appears to exist in a series of separate degrees, 
each one more refined and subtle than the preceding 
one and requiring a more highly developed species of 
consciousness for its comprehension. In other words, 
that intuitional glimpse of the essential character of 
reality, as viewed by the human consciousness, which 
impinged upon the minds of Riemann and Beltrami 
leading them to postulate as a corollary proposition 
to space-curvature, its manifoldness, is nothing more 
nor less than the intuition that the universum of spa- 
tiality cannot otherwise present itself to the intellect, 
owing to its peculiar adaptation to the sensuous, ex- 
cept by a series of continuous degrees which are per- 
ceptible only in proportion as the understanding is 
magnified to conform with it. After all, however, it 
is not improbable that the very objectivism of the 
universe in manifestation subsists in just the manner 
in which this intuitive glimpse implies and that the 
wisdom and utilitariness of the kosmogonic process 
which engendered spatiality are clearly demonstrated 
in that arrangement of the contents of the kosmos 
which presents the grossest elements of phenomena 



318 THE MYSTERY OF SPACE 

first to the intellect in its most impotent state while 
reserving the less crass for that time when the 
Thinker shall have evolved a cognitive organ adapt- 
able to its presentations. Those metageometricians 
who cling to the idea of the manifoldness of space, 
based as we have shown upon the pseudo-interpretation 
of a rather vague hint arising out of an unquestionably 
true intuition, have allowed themselves to fall into the 
unconscious error of magnifying the importance of 
the mere insinuation as to the space-nature to such an 
extent as wholly to obscure in their own minds and 
in the minds of those who think after them whatever 
of the true vision that may have been grasped by 
them. Furthermore, it is indubitably true that that 
same peculiarity of arrangement by which impalpable 
and invisible forces really subtend gross matter pro- 
ducing that subtle schematism in virtue of which the 
visible is subjoined to the invisible, the sensuous to 
the non-sensuous, spirit to matter, etc., also char- 
acterizes the appearance of spatiality to the human 
understanding. While there is a superficial semblance 
of separate and discrete manifolds into which space 
may be divided there are, in reality, no such sharp 
lines of demarkation between the subtle and the 
gross, between the visible and the invisible or between 
spirit and matter, each of these being capable of re- 
duction, by insensible degrees, into the other regard- 
less as to whether the reductional process originates 
on the side of the most refined or on that of the 
grossest. Accordingly, there ar£ no reasonable 
grounds upon which the notion of a space-manifold 
may be justified except as a metageometrical near- 
truth. 



METAGEOMETRICAL NEAR-TRUTHS 319 

In addition to the foregoing, there are yet other 
very fundamental considerations which would seem 
to debar the totality of analytical conclusions as to 
the nature of space from any claim to ultimate re- 
liability and trustworthiness. These are first: the fact 
that analyses are absolutely incapable of dealing with 
life; that being the direct product of a sort of 
mechanical consistency which marks the intellectual 
operations it has adaptability only for dealing with 
fragments or disconnected parts, and that without any 
reference whatsoever to the current of life or the 
flow of reality which has produced the parts. This 
fact is clearly shown in that attitude of the under- 
standing which inevitably leads it to the declaration 
that a line is an infinite series of points, a plane an 
infinite series of lines, and a cube, an infinite series of 
planes, and so on, indefinitely. To do this, to look 
upon all phenomena as a series of parts similar to 
each other and piled, one upon the other, or juxtaposed 
in the manner which they are discovered in the sen- 
sible world, is the natural tendency of the intellect 
and this tendency finds its most facile expression in 
analytics. Inadaptability of this sort is especially ob- 
servable in all problems of arithmetical analysis in 
which the vital element is a factor. When these 
analyses are carried to their logical conclusion, as has 
been shown in the chapter on "The Fourth Dimen- 
sion," invariably they end in an evident absurdity. 
But it is at their very conclusion where the life-element 
is encountered, where reality is approached, that they 
break down. The failure of analysis, then, to en- 
compass life, to fit into its requirements and to satisfy 
its natural outcome seems clearly to establish the basis 



3 2o THE MYSTERY OF SPACE 

of the perisophlcal nature of the entirety of analytical 
claims, especially that species of analysis which seeks 
the remoter fields of the conceptual for its determina- 
tions. Second: the close connection which has been 
seen to subsist between space and life as joint products 
of the same movement makes it obvious that the same 
ultimate rule of interpretation must be applied to 
both in. order to insure correct and dependable judg- 
ments regarding them. How different would be the in- 
tellectual attitude towards space if it were considered 
in the same light as vitality, provided one really un- 
derstood anything about vitality! Moreover, as it 
appears certain that the path of the intellect does not 
run in the same direction as the path which life makes, 
but in an inverse direction, it is clear that the judg- 
ments of the former, as to the action and essence of 
the latter, must necessarily be ultimately unreliable. 
It can readily be seen, however, that should the in- 
tellect be focused so as to follow the path of life, to 
attach itself to the very stream of life, it would have 
necessarily to neglect materiality. And such an ad- 
justment would, of course, obviate the need of a 
material life at all for humanity. In fact, a physical 
life with an intellect would be impossible under such 
conditions. It is well to recognize the suitability of 
the present schematism and not to become unwisely 
restive because of it; but it is also fitting that we should 
discriminate between that which is possible for the 
intellect chained to materiality and that which is impos- 
sible for it, in such a state, when foraying in a territory 
foreign to its nature, and beyond its powers to master. 
The predominating tendency in the intellect to 
account for the universe of life, mind and matter upon 



METAGEOMETRICAL NEAR-TRUTHS 321 

a strictly mechanical basis is undoubtedly due to the 
constitution of the intellect which does not admit of 
its direct consideration of the vital essence of things. 
We are bound ineluctably to the surface of things. 
All our knowledge is therefore superficial. We are 
even bound to the surface of ideas, and cannot pene- 
trate to the interior of these realities. Our art is the 
reproductions of superfices; our philosophies are the 
husks of eternahties; our religions, the habiliments of 
relations, and while it cannot be doubted that this 
arrangement is pre-eminently the best possible one for 
the present stage of man's evolution, it is nevertheless 
worth while to note that it is this very restricted 
activity of the intellect which shuts out from man's 
consciousness those very elements about which he is 
most concerned when he goes into the field of phi- 
losophy in search of a solution to his unanswerable 
queries. But some progress most surely is made when 
the mind is enabled to see its plight and recognize what 
are the difficulties and limitations that lie in its path 
of ultimate attainment. 

It is believed that the mechanistic, or true, char- 
acter of the intellect reached its zenith in the mind of 
Lagrange when he succeeded in reducing the entirety 
of physics to certain mechanical laws and formulae 
which he embodied in his "Mecanique Analytique" 
This work is undoubtedly the capstone of intellectual 
endeavors and stands as a monument which marks 
the culmination of the present stage of intellectual 
development. In thus placing the Mecanique at the 
apex of intellectual endeavors it is not thereby meant 
to be implied that the intellect shall not make more 
progress nor that other formulae, equally as marvel- 



322 THE MYSTERY OF SPACE 

ous as those which Lagrange discovered, may be 
devised, nor that other laws, heretofore undreamed 
of, may be found; but what is maintained is the fact 
that while there will be growth and development these 
will run along other channels, perhaps in the realm 
of the intuitable, and not any longer, especially so 
notably as now, in an opposite direction against the 
current of life and reality; and further, that there 
will be a gradual turning away from mechanics to 
biogenetics, from diversity to unity, from the purely 
intellectual to the intuitional, and withal a final getting 
rid of the bonds of illusion, of that thralldom of 
mechanics, whereupon will slowly arise the obso- 
lescence of all those disparities which may now be 
recognized in our knowledge and in the applications 
of the intellect to the data of the objective world. 

Because the intellect is unsuited to deal with real- 
ity, and because of its peculiar adaptation for diversity, 
for multiplicity, due to its mechanistic modus vivendi, 
there has grown up a voluminous catalogue of systems 
of philosophy. These embody such a multitudinous 
array of beliefs, ideas, conceptions, theories and con- 
jectures and constitute a movement in human thought 
which oscillates between the empiricist on the one 
hand and the transcendentalist on the other; between 
the idealist and the realist, leaning sometimes towards 
the Platonic, the Cartesian and the Kantian and at 
other times towards Spinoza, Aristotle, Spencer 
and Socrates, always terminating by multiplying the 
number of diverse beliefs rather than unifying them 
that the conclusion is unavoidable that so marked a 
lack of unanimity is indicative of a profound mental 
prestriction. It was, therefore, inevitable that mathe- 



METAGEOMETRICAL NEAR-TRUTHS 3*3 

matics should fall under the same spell and brook 
no let nor hindrance until it had succeeded in devising 
several diverse systems of geometry which it has done 
for the mere joy of doing something, of following its 
instinctive aptitudes. There is no other basis for the 
heterogeneity of our philosophies, our mathematics, 
indeed our beliefs than this mechanical, and hence, 
radically illusionary character of the intellect in con- 
sequence of which we have had to be satisfied with 
mere glimpses, hints, intimations and faint glimmer- 
ings of reality, of life, and of those kosmic movements 
which, if we had the ability to trace them from their 
source outward, would lead us unerringly to a truer 
and deeper knowledge of those things that under the 
present schematism must remain for us a closed book. 
The criterion of truth for us, constituted as we 
are and wedged in between the stream of life and its 
shore of materiality, must be that which relates our 
knowledge both to the stream and to the shore. It 
must be so that all predicates which purport to ap- 
proach it shall exhibit a dual reference — one that re- 
lates to materiality and another that relates to vitality, 
and yet a third that shall combine these two relations 
into one. All assertions, therefore, which pertain ex- 
clusively to either of these elements — to materiality 
or to life — are necessarily partial, fragmentary and 
perisophical in nature. Mathematics, because it re- 
lates to matter and the mechanical forces set up by 
matter acting against matter cannot be said to agree 
with such a criterion; art, because it relates to snap- 
shots or static views of matter is even more remote 
in its agreement; philosophy, as it has been known in 
the past and is known to-day, because it seeks to deal 



324 THE MYSTERY OF SPACE 

with a vitality fashioned after the image of mate- 
riality has failed when posited alongside of this 
criterion; and thus, the intellectual toil of millions of 
years has been in vain in so far as it has not succeeded 
even in raising a corner of the cover which hides 
reality from our view. 

A near-truth is any variation from this standard, 
this norm or criterion. It may be either logical, 
cognitive, scientific or even metaphysical. To define: 
a logical truth is a predicate based upon and involving 
the coherency and consistency of thoughts themselves; 
a cognitive truth is the conformity of knowledge with 
so much of reality as is known; scientific truth is the 
conformity of thoughts to things and conditions. All 
of these are obviously near-truths. Then, too, a near- 
truth may be defined as an assertion based upon the 
criterion of truth but falling within the category of 
cognitive truths owing to insufficiency of data or vision. 
Such indeed are those metageometrical predicates — 
w-dimensionality, space-flexure, space-manifoldness and 
all other assertions based upon these in general and 
specifically. Any recognition of truth must clearly 
embrace both the vital and material aspects of its 
subject in order to be adequately inclusive, that is, it 
should include the causative, the sustentative, the rela- 
tional and the developmental factors. These four 
factors are considered necessary and sufficient to de- 
termine the conformity of any view to the criterion of 
truth for when we are cognizant of the cause of a 
subject, understand the sustentative factors which keep 
it in existence, are conversant with its relations to other 
subjects and can follow its developmental variations 
until we come to its final status, why then, our knowl- 



METAGEOMETRICAL NEAR-TRUTHS 325 

edge is both sufficient and ultimate so far as that 
subject is concerned. Is it asking too much of mathe- 
matics or of philosophy or any system of thought that 
it conform to these standards or to this criterion before 
we shall accept it as final? Or shall we be satisfied 
with less than this? Let us hope not. 

In the foregoing presentation stress has been 
placed upon the fragmentary, and therefore, illusion- 
ary character of the intellect in order to arrive at an 
understanding of the difficulties under which real 
knowledge has to be acquired and to indicate the in- 
anity of all attempts to resolve the riddle of space 
by means of mathematics though regarded as the most 
typical exemplification of the mechanistic nature of 
the intellect. And further, to show that, on account 
of the radical incongruity which estranges life, the 
producer of spatiality, from the intellect which returns 
again to scrutinize the passage of life in its outward 
expressions, no hope of ever gaining the true view- 
point by means of the intellect need be entertained. 
But in doing so, it is deemed fitting that a note of 
warning should be sounded against any abortive at- 
tempts that may be made to obscure or distort the 
results of such a close discrimination lest the true 
import of the examination be lost for, if we emphasize 
the vanity of the intellect in the pursuit of that which 
it is by nature unsuited to attain we also equally stress 
the wise utilitarianism which limits it to the perform- 
ance of the tasks assigned while at the same time 
reserving for the function of more highly evolved 
powers, and indeed, for the intuition itself, the solu- 
tion of the riddle of spatiality. And if we declare the 
futility of the mathematical method in all endeavors 



326 THE MYSTERY OF SPACE 

aimed at unveiling the mysteries of life and mind, and 
of that movement which has its roots set in eternal 
duration from which it proceeds in an endless con- 
tinuity of purpose and promise, we do also recognize 
that in the science of mathematics the intellect shall, 
as in no other method of cognition, most fully fulfill 
the kosmic intent of its existence; and moreover, in 
the pursuit thereof it shall push the frontiers of its 
possibilities outward until it can be said almost to be 
able to make disposition of life itself — at least to 
that point where, when the intuition shall have come 
into its own, the passage from the mechanics of matter 
to the dynamics of life, shall be comparatively easy 
and natural. 



CHAPTER X 

The Media of New Perceptive Faculties 

The Spiritualization of Matter the End of Evolution — Sequence 
and Design in the Evolution of Human Faculties — The 
Upspringing Intuition — Evidences of Supernormal Powers 
of Perception and the Possibility of Attainment — The In- 
fluence and Place of the Pituitary Body and the Pineal 
Gland in the Evolution of Additional Faculties — The 
Skeptical Attitude of Empirical Science and the Need 
for a More Liberal Posture — The General Results of 
Pituitarial Awakening Upon Man and the Theory of 
Knowledge. 

Evolution is a continuous process and the primal 
impetus back of the great on-flowing ocean of life acts 
infinitively. It is not terminated when life has suc- 
ceeded in perfecting a form for the perfection of 
forms in themselves is not the end of vital activity. 
The end of evolution is the complete spiritualization 
of matter. So that it does not matter how perfect 
a form may be either subjectively or in its adaptation 
to environments; it does not matter how faultless a 
medium for the ensouling life it may be, there is ever 
the eternal necessity that life must drive it back over 
the path of its genesis until it shall be transmuted into 
pure spirit. Adaptation succeeds adaptation and with 
each there is a change in the form and this process 
continues until there is a more or less perfect con- 

327 



328 THE MYSTERY OF SPACE 

gruence between form and juxtaposed environmental 
conditions. But no sooner than agreement has been 
attained under one set of conditions new conditions 
arise and require a new setting, new adaptative move- 
ments. Thus there is a continuous proceeding from 
stage to stage, going from the grossest to the subtlest 
and most refined, always the form is being pushed 
onward and upward by life. But adaptation is not 
undergone for the benefit of the form, but more truly 
for the informing principle. It is the progression of 
the life-element which constitutes the adaptation of 
form to form and to their peculiar environs. The 
form is a tool or instrument of life which it discards 
the moment it fails to respond to its requirements. 
Thus forms are constantly being assumed and as con- 
stantly being relinquished. But no effort of life is lost 
regardless as to whether the action is performed in 
one or another form. The totality of matter is per- 
petually being acted upon by the totality of life. Every 
appulse of life against matter means an added push 
in the direction of spiritualization. The totality of 
such appulses of life against matter may seem infinitely 
small in the visible results which they produce in the 
process of spiritualization; but with each there is an 
eternal gain in that movement that shall end in the 
complete transmutation of materiality into spirituality. 
This action of life in metamorphosing matter, the 
nether pole of the great pair of opposites, into spiritu- 
ality, its copolar factor, in its outward, visible effects, 
is what we vaguely call evolution. J And such it is; 
for life is merely unfolding that which it has enfolded. 
Matter, having been involved as a phase of kosmic 
involution, is now being evolved. 



NEW PERCEPTIVE FACULTIES 329 

In the genesis of the kosmos there appear to be 
three great undulations in the universal current of life. 
The first of these prepares the field by depositing that 
elemental essence which is to become the world-plasm; 
the second precipitates the universum of materiality, 
spatiality and intellectuality, not as we now know 
them, of course, but as potencies; the third great un- 
dulation in the current of life effects the endowment 
of the world-plasm with those tendencies that are to 
build around themselves forms appropriate to their 
fulfillment. This ensoulment of the world-plasm with 
tendencies and the consequent segmentation of it into 
separate forms by these tendencies constitute the 
primary stages of that procedure of life which results 
ultimately in the up-raisement of matter and its final 
exaltation into pure spirit. Hence, the entire mass of 
materiality is besieged on all sides by the sum-total 
of life and the former is being raised slowly and 
irresistibly to heights that are immeasurably more sub- 
lime than its present degree of grossness. 

It appears paradoxical, therefore, that life, al- 
though in all respects vastly superior to matter, should 
become the apparent vassal of materiality and give 
itself up to all the strict rules of imprisonment which 
are imposed upon it by the properties and qualities 
which we observe in matter. It seems so subject 
to every whim and fancy of matter that one is in- 
clined to think that matter and not life is the chief 
designer of universal destiny. This is not a con- 
dition to be wondered at so much, for the reason 
that this apparent vassalage, this seeming enslave- 
ment of life by matter, is due to that superior and 
most marvelous adaptability of life which it enjoys 



330 THE MYSTERY OF SPACE 

in contradistinction to the relative unpliability of 
matter, and due also to the fact that life is kinetic and 
matter, being a mere deposit of life, is static. Life 
is mobility while matter is immobility and thus in 
possessing a greater range of freedom is, of course, 
correspondingly superior; but in this adaptation of 
itself to the labyrinthine cavities and multi formed in- 
terstices in matter it exhibits but a seeming serfdom 
which is really not a serfdom but a mastery. It is 
as if a man had taken lumber, hardware and stone 
and built a house wherein he might dwell — life has 
merely used matter, molded and fashioned it so as 
to make for itself a medium, a dwelling-place wherein 
it operates, not as a slave but as a master possessing 
unlimited freedom of motility. In the production of 
a form life stamps upon it, once for all time, the path 
of its engendering action. It leaves its finger-prints 
upon the mold which it makes for itself. So that if 
we would know where life has been or where it is we 
should look for its finger-prints (organization) ; we 
should observe the sinuosities which mark its path- 
way, remembering always that it is life that has formed 
the intricacies and complexities of the form into which 
it pours itself so accommodatingly in order that it may 
raise that form, develop and transmute it into some- 
thing higher and better. 

When we speak of form it must not be understood 
thereby that reference is made only to the gross 
physical form, but to the entire range of vital assump- 
tions or vehicles which life ensouls for purposes of 
manifestation. This range we believe to cover the 
whole path of kosmogenesis seriating from the densest 
to the most subtle. Our chief concern, however, is 



NEW PERCEPTIVE FACULTIES 331 

the immediate effect which the totality of life's opera- 
tions will have upon humanity or the form which it 
ensouls as the human organism. For it is impossible 
that humanity shall escape either the general or the 
specific results of the exalting power which life exerts 
over materiality and its appurtenances. It is, of 
course, impossible here to go into the various implica- 
tions of this general forward movement of the uni- 
versum of materiality or even to outline briefly the 
divergent lines of operation into which a satisfactory 
exposition of this view would naturally lead. And 
then to do so would be inappropriate in a volume of 
this kind. So we shall have to be content at this 
juncture to limit our study to a consideration of what 
we believe to be some of the immediate indications of 
this vast and most far-reaching phenomenon. 

In the chapter on the "Genesis and Nature of 
Space" it is shown that the material universe is en- 
gendered at the same time and by the same move- 
ment or process as the universum of spatiality and 
intellectuality and that as the passage from chaos to 
kosmos proceeds the function of this movement is 
changed gradually from engenderment to exaltation 
wherein materiality is transmuted into spirituality. It 
is, of course, obvious that as materiality is exalted 
so are spatiality and intellectuality; and that as the 
one becomes more and more refined, capable of an- 
swering to higher and yet higher requirements so do 
all the others. For, at work in all and through all of 
these, is the current of life which pervades them, en- 
gendering, sustaining and elevating as it proceeds. 
So that as matter has evolved added characteristics 
and properties, each answering to a given need and 



332 THE MYSTERY OF SPACE 

arising out of the necessities inhering in the stage at 
which it appeared, so has the intellect evolved facul- 
ties to correspond therewith. In other words, the evo- 
lution of faculties for the expression of the human 
intellect has proceeded synchronously with the evolu- 
tion of material qualities. And whenever a new 
faculty or an additional scope of motility is achieved 
by humanity there is always found a set of kosmic 
conditions which answers thereto. The cardinal prin- 
ciples of the doctrine of evolution are not, therefore, 
adverse to the conclusion that the organs of sense- 
perception — hearing, touch, sight, taste and smell — 
have not been endowed upon the human race or at- 
tained by it at one time; but rather that each answer- 
ing to a newly acquired need and opening a wider 
scope of motility for the intellect has been evolved 
separately and in due order. It would also seem that 
the quality of consciousness, as it has been manifested 
in the various stages of life through which it has passed, 
and especially the mineral, vegetal and lower animal, 
has not always been of the same degree of efficiency. 
Nor has it enjoyed the same kind of freedom which 
it now enjoys in the highly evolved genus homo. It 
is equally apparent that matter itself has not always 
been in possession of the same qualities and char- 
acteristics which it now exhibits; but that it, too, has 
gone through various stages of evolution bringing 
forward into each new stage the transmutfed results of 
each preceding one as a basis for further evolution 
and expansion. The innumerable archaeological evi- 
dences which support this view make it unnecessary 
to do more than state the facts, as they appear to 
be substantiated by indubitable testimonies. Further- 



NEW PERCEPTIVE FACULTIES 333 

more, it is believed that the outstanding implications 
of these phenomena will not be successfully contro- 
verted by those who are disinclined to see such implica- 
tions in the evolutionary process. In a previous 
chapter we have briefly sketched the characteristics 
which mark the upspringing of a new faculty showing 
how, at first, it appears as an abnormality which ex- 
hibits itself in a very few individuals only, and that 
in a more or less indefinite manner; and how later 
the number of individuals in which it appears gradually 
increases, the definiteness of the faculty, at the same 
time, appearing more marked; then, like a tidal wave, 
it recurs in a still larger number of persons until, at 
last after a long period of time usually several thou- 
sands of years, it becomes universal exhibiting itself 
in every individual and appearing as a hereditary 
characteristic of the entire human race. It is, there- 
fore, not without assurance as to the ultimate sound- 
ness of this view that we make the assertions which 
follow this brief introduction. 

It has already been stated that for a very obvious 
reason, namely, the satisfaction of the needs of our 
present humanity, the intuition is for the time dom- 
inated by the intellect and held in subjugation by it so 
that all of man's external operations are governed 
and dictated almost entirely by the intellectuality, al- 
lowing the intuition only rare moments when it can 
come to the fore at all. This is the rule in the evolu- 
tion of faculties and characteristics. The higher 
faculty, although potentially present in every way, is 
ever held in abeyance while the lower is brought, under 
the rigors of its own evolution, to a point where its 
joint operation with the higher may be executed with 



334 THE MYSTERY OF SPACE 

the least possible friction and retardation as also with 
the greatest possible coordination and cooperation. 
Accordingly, notwithstanding the fact that materiality 
must possess in potentiality all the qualities which it 
will at any time reveal, it is nevertheless necessary that 
these qualities shall come forth gradually and in due 
order. Similarly, humanity has come into possession 
of its various faculties of mind, and powers of physio- 
logical functions, by insensible degrees, the higher 
always being held in abeyance until the lower is fully 
developed. Those faculties which are to bestow 
added powers, additional freedom and a greater scope 
of motility are the ones which appear later than those 
which are truly primitive in character. These facts 
have been amply demonstrated by the science of em- 
bryology wherein it is shown that ontogeny is a re- 
capitulation of phylogeny. That is, the history of the 
development of the individual is a recapitulation of the 
development of the species. Thus the various stages 
of development through which the human embryo 
passes while in utero are but a repetition of similar 
stages through which the entire human species has 
passed in its phylogenetic development. Wherefore, 
it is certain that humanity has not attained, at one and 
the same time, all the powers of mind and body which 
it now possesses; that the childhood of the human race 
represented a time when it had but few faculties or 
organs of sense-perception — indeed a time when the 
higher sense-organs of smell, taste and sight were 
entirely lacking although residing in potentiality 
therein. 

It is undoubtedly true that the earth has passed 
through a similar evolution with respect to its own 



NEW PERCEPTIVE FACULTIES 335 

material characteristics, that its childhood was, in all 
points, analogous to the childhood of humanity; that 
the air, earth and water were wholly absent, except 
in potentiality, during the nebulous youth of its genesis. 
It is even probable that there are at work to-day 
processes which in the future shall culminate in the 
evolution of newer, higher and more complicately or- 
ganized species of plants, animals and minerals. Every 
year brings fresh evidences that crystallize the con- 
viction that the earth has been the scene for the ap- 
pearance of many strange orders of animal life. 
Fossiliferous strata are continually yielding incontest- 
able testimonies of changing flora and fauna. We 
count the animal and vegetal life of to-day as being 
more highly developed than that of any other previous 
age, and it is well that this is so, for simplicity of 
organization and primality of manifestation are always 
succeeded by complexity and a greater scope of 
adaptability. 

We have said that the whole of that movement 
of the intellect which has brought forth the meta- 
geometrical creations of hyperspaces, the curvature of 
space and its manifoldness together with the entire 
assemblage of mathetic contrivances are merely the 
early evidences of the appearance in the human race 
of a new faculty, a new medium of perception whereby 
the Thinker shall acquire a still greater range of 
motility than that now offered by the intellect. Atten- 
tion has been called also to the fact that this phenom- 
enon has been manifested not alone in the field of 
mathematics, but in art, religion, politics and also in 
science in which we have only to witness the marvelous 
strides already made in the discovery of radio-active 



336 THE MYSTERY OF SPACE 

substances, the Roentgen, Becquerel, Leonard 
and other kinds of rays. It is quite confidently be- 
lieved that these forward movements in every branch 
of intellectual pursuit, these combined efforts of the 
intellect, in peering into the occult side of material 
things, are in response to the evolutionary needs of 
the Thinker, and in addition, are the evidences, and 
shall in time be the cause, of the development of an 
additional set of faculties. Function, or the perform- 
ance of acts, determines faculty or the power of action 
and ultimately the organ itself. Thus the mere wish 
to perform aroused by desire and vitalized by the 
will actually terminates, in the course of time, in the 
genesis of a faculty, or the power to perform. The 
constant upreaching yearnings of the Thinker through 
his intellect for greater freedom and a larger scope 
of action, the desire to peer into the mysteries of life 
and mind, the infantile out-feelings of the mentality 
after some safer and surer basis for its theory of 
knowledge cannot fail in producing not only the 
faculty or power to satisfy these cravings but the very 
organ or medium by virtue of which the satisfaction 
may be attained. 

It is not strange that in mathematics the intellect 
should have found first the clue to the existence of 
a higher sphere of intellectual research wherein it 
might become the creator of the various entities which 
peopled the new found domain; it is not strange that 
the mathematician should, in this instance, have as- 
sumed the role of the prophet proclaiming by various 
mathetic contrivances (although unconsciously) that 
the human race is nearing that time when it shall 
actually be able to function consciously in some higher 



NEW PERCEPTIVE FACULTIES 337 

sphere; neither is it to be wondered at that the voice 
of the prophet is heard and respected throughout the 
earth; for, indeed the mathematician is a spokesman 
who, as a rule, is unmoved by sudden outbursts of 
passion and ecstatic frenzies of emotions but calmly 
and dispassionately verifies his conclusions, tests them 
for consistency and having found them to satisfy the 
most rigorous mathetic requirements hesitates not to 
propound them. For this cause humanity respects the 
mathematician, and when he speaks listens to his voice. 
It is well, too, that this is so; for the history of mathe- 
matics is clearly the history of the development of 
the intellect. So exact a determinator of the quality 
of intellectual efficiency is it that its reign may be 
said to be an absolute monarchism whose lines of 
dominance extend to the minutest desire or appetency. 
It has always been the guide of the intellect, going 
before, as it were, blazing the trail, pushing back the 
frontiers of the intellect's domain and clearing away 
the debris so that the intellect with its retinue of 
servitors might have an easy path of progress. 

Mathematics, however, has not the aptitude to 
serve the intuition as it serves the intellect. So the 
path into which the intuition would lead humanity the 
mathematician, because of his training and peculiar 
functions, is unprepared to enter. It is for this reason 
that when mathematics leads the intellect up to that 
point where it encounters life it fails, it becomes 
confused and its dictatorship becomes a mockery, its 
decrees remain unexecuted and futile. In taking this 
view we have certainly no desire to offend the mathe- 
matician or to detract from the glory of his mon- 
archistic rulership over the intellectual progress of the 



338 THE MYSTERY OF SPACE 

race; for, in truth, mathematics is the diadem of gold 
wherewith man has crowned his intellect. Yet it is 
well, yea imperative in the light of recent developments 
in the realm of hyperspace, that a careful discrimina- 
tion should be made as between the sphere of the 
intellect and that to which the intuition shall attain. 

The intuition, long held in abeyance until the in- 
tellect should be fully crowned and reach the zenith 
of its powers, is now coming to the front. It will be 
many centuries perhaps before it shall have grown to 
such proportions as those already attained by the in- 
tellect; perhaps a few thousand years may pass before 
the intuition shall have evolved to that point where 
it may labor as coadjutor to the intellect; but un- 
doubtedly the time will come when it, too, shall reward 
the Thinker's labors with that which shall be more 
precious than the crown of gold which the intellect 
has won. Then, the intellect, grown old and decrepit 
with years of reigning shall become dim and crystal- 
shaped and finally pass into automatism or reflexive 
movements where without the urge of volitional im- 
pulses it will perform with exactness, precision and 
utter loyalty the tasks which it has learned so well 
to execute in the days of its forgotten glory. Man- 
kind will then be free. A new freedom, wherein the 
erstwhile lightning flashes of intuition will become 
fused into one glorious sheen of all-revealing light, 
shall come to men and thus the race resplendent will 
walk the earth enshrined in the majesty of divine 
powers attained as a result of millions of years of 
aspiration. 

That there are supersensuous realms so far above 
the range of our senses as to be entirely beyond their 



NEW PERCEPTIVE FACULTIES 339 

ken needs now no proof or argument; for the scientist 
has demonstrated, by the invention of instruments of 
extreme delicacy and precision, that such a world 
does really exist. Already we know of stars so dis- 
tant that, though light traverses in the brief space of 
an hour six hundred million miles, they might have 
ceased to shine before the pyramids were built and 
yet be visible to us in the skies. If the human eye 
were as sensitive as the spectroscope many thousand 
tints and shades might be added to the world of 
color; if they possessed the magnifying powers of the 
microscope we should live in constant terror and awe 
of the monstrous entities that teem in the water which 
w T e drink and in the air which we breathe; and if our 
ears could detect the microphonic vibrations which 
register in the delicate apparatus of some microphones 
the dead, vacuum-stillness of nature's great silences 
would appear as a babel of voices by the seaside. 
The sense of touch, responding to the same range of 
vibrations as the micrometer, would reveal actually the 
interstices between particles of the densest elements; 
and gold, silver, platinum and mercury would seem 
but honeycombs of matter. But, to the forward- 
looking there is no element of absurdity in the ex- 
pectation that all these senses shall, one day, be able 
to dispense with the artificial aid of physical ap- 
paratus and perform, with even greater precision and 
faithfulness, the task which they now perform so 
crudely and ineffectively. There are without doubt 
vibrations of taste and smell which are so far above 
the range of these senses that they have no effect upon 
them whatsoever. Notwithstanding the fact, however, 
that the galvanometer, microscope, the microphone, 



34° THE MYSTERY OF SPACE 

the spectroscope and the telescope have extended thus 
the sphere of sense-knowledge there are yet subtler 
vibrations to which these delicate instruments do not 
and ought not be expected to respond. But to say, 
as do many empiricists, that since these phenomena 
cannot be detected by scientific instruments they do 
not, therefore, exist seems to be expecting too much 
of material means as well as exposing oneself un- 
necessarily to criticism on the grounds of extreme 
materialistic appetences. 

There is indeed need of a more liberal attitude 
among men of science towards the world of the un- 
seen. Intolerance of the data which it offers will for 
a time perhaps preserve the aloofness of scientific 
dogmatism inviolate but there will most surely come 
a reaction against the dogmatism of science and men 
will seek freedom and attain it despite their fetters. 
Sir Oliver Lodge, in his book, the Survival of Man, 1 
says: "Man's outlook upon the universe is entering 
upon a new phase. Simultaneously with the beginning 
of a revolutionary increase in his powers of physical 
locomotion — which will soon be extended to a third 
dimension and no longer limited to a solid or liquid 
surface — his power of reciprocal mental intercourse is 
also in process of being enlarged; for there are signs 
that it will some day be no longer limited to con- 
temporary denizens of earth, but will permit a utiliza- 
tion of knowledge and powers superior to his own, 
even to the extent of ultimately attaining trustworthy 
information concerning other conditions of existence." 

It is the author's good fortune that he has for a 
period extending over several years been able to verify 
1 See pp. 338, 341. 



NEW PERCEPTIVE FACULTIES 341 

the conclusions which Sir Oliver Lodge expresses in 
the above, and thus to satisfy his own mind that the 
process by which man's mental powers are "being en- 
larged" is indeed demonstrable by actual observation 
and experimental methods. 
Lodge continues: 

"The boundary between the two states — the 
known and the unknown, is still substantial, but it 
is wearing thin in places, and like excavators en- 
gaged in boring a tunnel from opposite ends amid 
the roar of water and other noises, we are beginning 
to hear now and again the strokes of the pick-axes 
of our comrades on the other side." 

Camille Flammarion 2 cites 436 cases of psy- 
chic manifestations examined by himself and which 
establish beyond any reasonable doubt that there are 
certain perceptive faculties, namely, clairvoyance and 
clairaudience, that crop out in certain individuals, 
sometimes in abnormal conditions and sometimes 
normally, the very unusual character of which proves 
their rudimentary nature and the potency of their 
maturescence in the humanity of the future. Among 
the cases cited by Flammarion are 186 instances of 
manifestations from the dying received by persons 
who were awake; 70 cases were manifestations re- 
ceived by persons asleep; 57 were observations of 
direct transmission of thought without the aid of sight, 
hearing or touch or other physical means; 49 were 
cases of sight at a distance or clairvoyance by per- 
sons awake, in dreams or in somnambulism and 74 
cases of premonitory dreams or predictions of the 
3 See Unknown, p. 485, et. seq. 



342 THE MYSTERY OF SPACE 

future. Indeed, there are few persons now living who 
have not had similar experiences, if not exactly like 
these, of the same nature. These examples, of course, 
may be greatly multiplied in every country in the 
world, and it is unnecessary to enumerate them 
further; for, when once the existence of such faculties 
has been demonstrated in persons, either in a normal 
or an abnormal condition, their presence can no longer 
be questioned by the fair-minded. It is, then, only a 
question of evolution before they will appear in the 
normal way and their universalization, as transmissi- 
ble characters, be an accomplished fact. When we 
are brought face to face with this sort of phenomenon 
which seems to be increasing rapidly the conclusion 
is inevitably forced upon us that since evolution must 
be a continuous process and matter destined to yield 
higher and more refined powers and humanity to come 
into a far more extensive scope of motility because 
of the opening avenues of knowledge, it is not im- 
possible that these acuter senses, these new faculties 
are now existing in the human race in a rudimentary 
stage and are designed to become the universal pos- 
session of all. That this is to be the almost immedi- 
ate outcome of the perpetual exalting power which 
life exercises not alone over materiality but over 
human organs and faculties as well, seems to be the 
one big, outstanding implication of the evolutionary 
process. The presence of such functions as the ability 
to sense the invisible and the inaudible, to answer to 
vibrations far subtler than anything in the scope of 
our external sense-organs, certainly indicates the ex- 
istence of rudimentary faculties which make these 
functions possible. Back of these vague, indefinite 



NEW PERCEPTIVE FACULTIES 343 

functions, back of every supernormal or abnormal 
manifestation of man's mentality and back of all that 
class of phenomena which take their rise out of super- 
sensuous areas must lie, in ever increasing potency, 
faculties and organs, however rudimentary, which are 
the source of these manifestations. Life, that in- 
eluctable agent of creation, which is incessantly push- 
ing outward the confines of the intellect's scope of 
motility, never wearying, never tiring nor sleeping, has 
long ago, in the dim and distant past of man's evolu- 
tion, laid the foundations; and in fact, with one stroke 
of its creative hand, has molded the organs which are 
to become the active media of these new faculties. 
And now, these incipient demonstrations, these in- 
fantile struggles which we see now and again out- 
putting from them, are but the specializing processes 
through which, in their later development, these 
organs are proceeding. These are the outward signs 
which should tell us that life is breaking up these 
organs into special parts, assigning to each a certain 
division of labor and making of each a perfect co- 
ordinate of all the others. It is, by these very dispread 
exhibitions, cutting up, specializing and by slow de- 
grees determining the function, character and general 
tendence of the organs of expression wherewith these 
manifestations shall be centralized and put into ef- 
fective operation. In doing this, it is but following 
its accustomed procedure, the procedure which it 
adopted when it produced the eye, the ear, the heart 
and the spleen. We shall, therefore, gauge our un- 
derstanding of the purport and end of evolution; in 
fact, we shall determine our exact intellectual com- 
prehension of life itself, by the attitude which we 



344 THE MYSTERY OF SPACE 

adopt towards it and the mode of its appearance. 
Much depends, accordingly, upon the posture which 
we assume towards life — whether we shall say the 
totality of life's creative powers has been dissipated 
in the bringing of the human body to its present 
degree of perfection; whether we shall say that it is 
neither necessary nor possible for life to produce 
other organs and faculties which shall respond to the 
unseen world about us revealing its glories in a way 
far more perfect than do our external sense-organs 
reveal the wonders of the world of sensation; or, 
whether we shall conclude from these most palpable 
evidences that life has yet other powers and faculties 
which it designs to bestow upon the human mind and 
other organs and capabilities with which it shall endow 
the human body so that man, in his evolution, shall 
be enabled to rise to still higher spheres while yet 
incarnate. There may be, and undoubtedly are, those 
who, for various reasons prefer to take the former 
positions and there are certainly those who like 
Lodge, Flammarion, Hudson, Crookes and a host 
of others, preferring the latter view, would rather 
believe in the strength of the great mass of corrobora- 
tive testimonies that we are even to-day in the midst 
of the matutinal hours of a newer, a better and a 
far more efficient era of human evolution than any 
through which we have hitherto come. 

Already, recent scientific investigations and the 
results obtained therefrom have begun to turn the at- 
tention of medical authorities to the activities of two 
very small organs situated in the mid-brain and known 
as the pineal gland and the pituitary body. These 
organs, and especially the pineal gland hitherto sup- 



NEW PERCEPTIVE FACULTIES 345 

posed to be a vestige of the past, are now beginning 
to be recognized as rudimentary organs belonging to 
the future evolution of humanity. Dr. Charles de 
M. Sajous, who is an authority on the pituitary body, 
believes that it has no active internal secretions but 
is an "epithelio-nervous organ" which controls, 
through nerves leading to the adrenals and thyroid 
bone, the processes of general oxygenation, metab- 
olism and nutrition. Little is known of the functions 
of the pineal gland except that it is an ovoid, reddish 
organ attached to the posterior cerebral commissure 
projecting downward and backward between the 
anterior pair of the corpora quadrigemina. It is 
otherwise known as the "conarium," the "pinus" or 
"epiphysis." Situated at the base of the brain, it is 
held in position by a fold of the pia mater while its 
base is connected with the cerebrum by two pedicles. 
It contains amylaceous and gritty, calcareous particles 
constituting the brain sand. There are, however, 
marked structural resemblances between the pineal 
gland and the pituitary body and their formation is 
perhaps the most interesting feature of the develop- 
ment of the thalamencephalon or mid-brain. The 
hypophysis cerebri or pituitary body is a small, ovoid, 
pale, reddish mass varying in weight from five to ten 
grains and situated at the basal extremity of the brain 
in a depression of the cranium known as the sella 
turcica, a configuration very much like a Turkish 
saddle in shape. It is a composite, ductless gland 
and consists of two divisions, an anterior and a 
posterior, connected by an intermedial portion — all of 
which are attached to the base of the cerebrum by the 
infundibulum. The anterior lobe is larger than the 



34 6 THE MYSTERY OF SPACE 

posterior and very vascular, springing in its develop- 
ment from the buccal cavity of the embryo; the pos- 
terior lobe is situated in a depression of the anterior and 
is a brain-process. The pituitary body itself is lodged 
in a cavity of the sphenoid bone called the pituitary 
fossa. This is a most remarkable position, for the 
reason that the sphenoid, or wedge-shaped, bone which 
lies at the base of the skull articulates from behind 
with the occipital and in front with the frontal and 
ethmoid bones and by lateral processes with the 
frontal, parietal and temporal bones. From this posi- 
tion it binds together all the bones of the cranium, 
and moreover, articulates with many bones of the face. 
It is upon the upper surface of the sphenoid bone 
which occupies such a prominent and commanding 
position in the cranium, in a deep depression, that the 
pituitary gland is located. 

Each nasal chamber is lined by a mucous mem- 
brane called the pituitary or Schneiderian. This mem- 
brane is prolonged into the meatuses and air sinuses 
which open into the nasal chambers. The pituitary 
membrance is thick and soft and diminishes the size 
of the meatuses and air sinuses. It is covered by a 
ciliated columnar epithelium and contains numerous 
racemose glands for the secretion of mucous or 
pituita. It is also vascular and the veins which ramify 
it have a plexiform or net-work like arrangement. It 
divides into two membranes — a respiratory, which is 
concerned in breathing, and an olfactory region. The 
respiratory region corresponds to the-floor of the nose, 
to the inferior turbinated bone and to the lower third 
of the nasal septum. The olfactory region is the seat 
and distribution of the olfactory nerve and corre- 



NEW PERCEPTIVE FACULTIES 347 

sponds to the base of the nose, to the superior and 
middle turbinals and the upper two-thirds of the nasal 
septum. 

Recent developments prove that this gland has a 
profound influence over the animal economy. It is 
believed by some that the pituitary body actually de- 
stroys certain substances which have a toxic influence 
on the nervous system; others believe that it secretes 
material media for the proper action of the trophic 
or nutritive apparatus; still others believe that it in- 
fluences blood-pressure. It is known, however, from 
experimentation, that its removal in dogs, cats, mice 
and guinea pigs causes a fall of temperature, lassitude, 
muscular twitchings, dyspnoea or difficult breathing, 
and even speedy death. Hypertrophy of the gland 
is directly associated with certain diseases, such as 
giantism and acromegaly. The latter is a disease 
which causes a general enlargement of the bones of 
the head, feet and hands, usually occurring between 
the ages of twenty and forty years, and most frequently 
in females. The fact that these diseases are so closely 
associated with a hypertrophic condition of the 
pituitary gland has led to the conclusion that perhaps 
the giants or Cyclops of ancient times were cases of 
giantism or acromegaly. This view, while interesting 
from the standpoint of the functions of the pituitary 
gland, is not necessarily a correct one; for the age 
of giants, when men attained to a much larger stature 
than at present, can be accounted for on other grounds, 
namely; that the early mesozoic man, on account of 
his having to live among animals, trees and other 
vegetation of such huge size, had naturally to be fitted 
with a frame proportional to other animals in order 



348 THE MYSTERY OF SPACE 

that he might successfully cope with his environing 
conditions. Nature thus wisely fitted him for the 
conditions which she had prepared for the scenes of 
his life. 

The facts adduced in the foregoing description are 
purely empirical and may be verified by any who seek 
to establish their correctness or incorrectness. But 
we are about to introduce a species of testimony which 
while it may also be verified will not be found so 
easy of verification as the above-mentioned physio- 
logical facts, and not by the same means; yet they 
are nevertheless deserving of a place here. It is the 
liberal attitude that we must adopt towards all 
phenomena, excluding none that give promise of the 
widening and deepening of our knowledge and an ex- 
planation of much that has seemed heretofore unac- 
countable. 

We have noted how subtle is the physical connec- 
tion between these two bodies, the pineal gland and 
the pituitary body; we have seen how profound is the 
effect which the latter has been demonstrated, in a 
measure, to have over the entire bodily economy; but 
there is even other testimony to the effect that those 
gifted with the inner vision can observe the "pulsating 
aura" in each body, a movement which is not unlike 
the pulsations of the heart and which never ceases 
throughout life. In the development of clairvoyance 
it is known that this motion becomes intensified, the 
auric vibrations becoming stronger and more pro- 
nounced. The pituitary body is the energizer of the 
pineal gland and, as its pulsating arc rises more and 
more until it contacts the pineal gland, it awakens and 
arouses it into a renewed activity in much the same 



NEW PERCEPTIVE FACULTIES 349 

manner as current electricity excites nervous tissue. 
When the pineal gland is thus aroused clairvoyant per- 
ception is said to become possible. These are facts 
which cannot be proved by the materialistic man of 
science nor can they be demonstrated to the layman 
who has to depend alone upon sense-deliveries for his 
knowledge. This is true for the reason that, in the 
first place, it is necessary that he shall either feel in 
his own mid-brain the energizing activity of these two 
organs and have his entire nerve-body shaken from 
crown to toe by the down rushing currents of that 
subtle energy with which the pituitary body floods it or 
be himself the perceiver of its activities. Nevertheless 
attention is here called to these phenomena and the 
conclusions drawn therefrom are offered as a means of 
denoting the probable line of investigations which will 
establish the directions which we should pursue and 
the source whence we shall find outcropping the new 
faculties and their organs of expression. 

We confess to a knowledge of the fact that men 
of empirical science have long maintained a rather 
skeptical, if not contemptuous, attitude towards all 
these phenomena but it is also felt that there is far 
more of discredit in their attitude than of credit; for, 
in so doing, they have voluntarily adopted measures 
by means of which the knowledge that they so eagerly 
seek is shut out from their attainment. In vain, then, 
is appeal made to the intellect to remove the bar- 
riers which it unconsciously interposes between itself 
and the goal of its pursuit; in vain do we appeal to 
the materialist to give ear to testimony the data of 
which cannot be made amenable to his knife and 
scalpel neither to the microscope nor microphone; in 



350 THE MYSTERY OF SPACE 

sheer vanity is he adjured to look within — into the in- 
terior of life, of mind and the things which he handles 
with his instruments — for the answers to his queries, 
for the path which leads into the wake of life and 
consciousness. Because his utter loyalty and devotion 
to the modus vivendi of the intellect will not permit 
this; but, after all, it is not wholly wise to allure him 
away unbetimes from his search after truth through 
superficialities nor to inveigle him into giving up his 
tenacious prosecution of the physically determinable. 
We would not have it so; for, perchance, he, too, 
one fine day, in the quiet of his laboratory shall come 
upon the data which may substantiate in his own mind 
the long settled conclusions of the occultist who, fre- 
quently and not without cause, grows impatient at the 
scientist's obstinate delay. These two workers, the 
empiricist and the occultist, must ultimately come to- 
gether as collaborators — the one working upon the 
form, the vehicle, physical matter and the other seek- 
ing to understand the life, the interior forces which 
produce, the creative element. They cannot remain 
always aloof from one another; for they, too, are as 
men digging a tunnel from opposite ends. Finally, 
the partition will break and thus will dawn a new day 
for the knowledge of humanity and men will see the 
rationale, the truth and good sense of cooperation 
in this respect. 

It can be said with confidence that whatever in the 
future may be learned as to the physiologic functions 
of the pituitary body and the pineal gland, it suffices to 
know that it is life which they express and that, too, 
in a far superior manner than any of the other sense 
organs. The modus of these two glands differs in a 



NEW PERCEPTIVE FACULTIES 351 

very marked way from that of the organs of sight, 
hearing, taste, smell and feeling. For these latter are 
designed for contact with the external, objective world 
of sensations, their growth and evolution being de- 
pendent upon stimuli received from without while with 
the former the case is far different, in fact, just the 
opposite. The mode of life of the pituitary body and 
the pineal gland, instead of receiving sustenance and 
impetus from external stimuli, is rather dependent upon 
impacts received from the Thinker's own Conscious- 
ness and made to impinge upon them by an exclusively 
interior process. Situated in the mid-brain, safely 
secluded from all external interference, they are 
naturally limited to stimuli which come from within, or 
it may be said, they are responsive to excitations that 
are more spiritual than those which come through the 
external sense-organs. If, as has been said they con- 
trol the internal processes of metabolism (anabolism 
and katabolism), oxygenation, nutrition, and other 
important internal movements, none of which can be 
said to be under the control of the intellect, is it 
not, therefore, justly assumed that their response is 
directed towards stimuli which arise interiorly or upon 
a plane higher than the intellectual? It is a matter 
of scientific knowledge that those persons gifted with 
clairvoyance, and commonly known as "sensitives" are 
far more responsive to nervous excitation than those 
who are not so gifted. This would seem to imply 
that, on account of the superactivity of these two 
organs, the entire nerve-body has, in consequence, be- 
come more delicately and subtly organized. They 
seem to act as a switchboard for the regulation of the 
flow of the current of life through the body. Not 



352 THE MYSTERY OF SPACE 

only do they come more nearly to an adequate ex- 
pression of the physiologic function of life, but, as 
their energization means an enlargement of the scope 
of perception by giving the Thinker's active conscious- 
ness access to hitherto unapproachable realities and by 
penetrating the outer mask which life ensouls and also 
laying bare a domain of unlimited knowledge the 
manifestation of which is far more real than anything 
the senses can disclose, it is evident that they constitute, 
in their collaborative functions, a more highly adapt- 
able medium for the expression of the Thinker's con- 
sciousness. And if so, for the kosmic consciousness 
which is the source of all forms of consciousness, they 
furnish a specializing and adaptizing agency. 

Now, in all those cases of inspirations, revelations, 
telepathic communications, clairaudience, clairvoyance, 
dreams, visions, etc., wherein the Thinker is enabled 
to perceive facts and verities which are then presented 
to his consciousness in a manner clearly without the 
province of the common sense-organs, it must be ap- 
parent that these manifestations are apprehended by 
a perceptual mechanism which is entirely independent 
of external sense presentations but which is an interior 
and subtler form of psychic activity. Sounds which 
are heard by so-called "sensitives" and objects which 
are perceived by eyes that are keener than those organs 
said to have been evolved from the "medusa" cannot 
be heard by other persons nor perceived by them in 
any way. Thus it would seem that there are inner 
organs of perception which respond to these finer 
vibrations and which enable the person so gifted to 
apprehend them. 

There are those who, presumably basing their 



NEW PERCEPTIVE FACULTIES 353 

assertions upon actual observation and knowledge, un- 
qualifiedly assert that in order "to gain contact with 
the inner worlds all that remains to be done is the 
awakening of the pituitary body and the pineal gland. 
When this is accomplished man will again possess 
the faculty of perception in the higher worlds, but 
on a grander scale than formerly (when humanity was 
in its infancy and exercised a lower form of psychic 
power only) ; because it will be in connection with the 
voluntary nervous system, and therefore, under the 
control of the will. Through this inner perceptive 
faculty all avenues of knowledge will be opened to 
him and he will have at his service a means of ac- 
quiring information compared with which all other 
methods of investigation are but child's play." 3 It is 
the lack of this ability to see, with our physical eyes, 
as it were, by the "Roentgen rays," to penetrate 
the inwardness of things that has baffled and con- 
founded men for so long a time and which has 
eventually led certain mathematicians and others to 
conjecture such strange, and in many cases, illogical 
possibilities for the denizens of four-space. This in- 
ability together with the desire to fathom the inner- 
most complexities of solids and to handle, albeit with 
unholy hands, the supersensuous, the mysterious and 
the unapproachable identity of "things-in- themselves" 
have induced the more zealous among them to con- 
trive some kind of hypothesis which would, at least, 
offer an explanation of these phenomena. It has 
driven them to wrestle with metaphysical possibilities 
in a vain endeavor to grasp that which, ignis-fatuus 
like, ever evades their slightest intellectual approach. 
3 Rosicrucian Cosmo-Conception, p. 477, Max Heindel. 



354 THE MYSTERY OF SPACE 

But why this prolonged struggle, why this intellectual 
maneuvering and sophistry? "We can calculate, 
compute, excogitate," says Paul Carus, 4 u and de- 
scribe all the characteristics of four-dimensional space, 
so long as we remain in the realm of abstract thought 
and do not venture to make use of our motility and 
execute our plans in an actualized construction of 
motion; but as soon as we make an a priori con- 
struction of the scope of our motility, we find out 
the incompatibility of the whole scheme." Thus 
mathematicians are forced to relinquish all hopes of 
transforming the world of life into a sort of four- 
space dwelling place where everything is done accord- 
ing to the laws of mathematics. But whether they 
shall accept it or not there is a wider, truer and more 
rational view which recognizes all metageometrical 
investigations, as well as all kindred phenomena, as 
universal evidences indeed, as the very causes which, 
in the future humanity, will actually awaken and 
cause to be accelerated in their development these 
little inner sense-organs, the pineal gland and the 
pituitary body, whose perfect development promises 
to provide for the Thinker's consciousness an avenue 
of expression such as humanity has possessed never 
before. And too, it is not without full knowledge of 
the fact that it has been customary, among certain 
scientists or perhaps all of them, to regard these 
bodies, at least the pineal gland, as vestigal organs 
belonging to the past of human evolution, that we 
make these assertions. Yet, as manproceeds in the 
perfection of mechanical science, in the development 
of instruments of precision that aid his external senses, 

* Foundations of Mathematics, p. 90. 



NEW PERCEPTIVE FACULTIES 355 

responds more and more to the subtle vibrations teem- 
ing everywhere in the atmosphere about him, and 
comes, in the course of time, naturally to possess a 
more sensitively keyed nervous mechanism, a finer 
body and higher spiritual aspirations, there will be 
a corresponding widening of his scope of vision and 
the attainment of larger powers of perception which 
must inevitably, in the very nature of things, tend 
towards a deeper and truer knowledge. 

In view of the foregoing, it is believed that the 
general results of this pituitarial awakening which 
may be expected as humanity continues to evolve 
should be seen in the marked effects which will be 
wrought in the entire metabolistic area of the human 
body whereby a gradual intensification and sensitiza- 
tion of the whole neural mechanism will raise the 
peculiar efficiency of all the senses whether purely 
physiologic or psychic. For there are undoubtedly 
notes so delicate in their intensity that they transcend 
the grasp of the audital nerves; scents and fragrances 
so subtle in their excelling purity that it is beyond the 
powers of our present olfactorial contrivances to de- 
tect them; colors and other external stimuli so sub- 
limely supersensuous that a nervous mechanism per- 
haps ten-fold more delicate and responsive than ours 
is required to apprehend them. All these, and more 
than at present is conceivable, will come, with the 
aid of pituitarial stimulation, within the purview of 
a more highly developed humanity of the future. And 
because mathematics have led a movement into the 
very camp of the intellectuals — logic-bound and 
tethered by the severest rigors of mathesis — whereby 
the intolerant intellect has been compelled, by rules 



356 THE MYSTERY OF SPACE 

of its own making, to recognize the existence of the 
supersensuous, and by looking into the glaring light 
of the sun of the intuitable to gain strength of vision 
and boldness to press forward, a great and far- 
reaching service has been wrought for humanity. And 
in the tower of hyperspace mathematics have erected 
a monument to the intellect which, as long as the 
human race remains, will mark the great turning point 
in man's path to the highest life. 

What if it were possible that the scientist, when 
he had carried instruments to their utmost precision 
and penetration, should suddenly, or otherwise, be 
endowed with a clear-perceptivity of sight, hearing and 
smell, so that he could with his own powers of vision, 
feeling and hearing take up the task where the micro- 
scope, the microphone and the micrometer left off 
and delve into depths far too unfathomable for his 
appliances, perceiving the innermost realities of 
things and processes? What if it were possible for 
him, with these added powers, to see and examine 
without the aid of the magnifying lens the electron, 
the atom and the molecule? What if the cell, the 
bacterium, and other invisible forms of life would 
then deliver up their secrets to his knowing mind? 
What if he could sense with his own inner vision, the 
ultra-violet and the infra-red rays; what indeed, if 
spirit itself, the innermost sheath of life, should be 
visible and palpable to him and he could note the 
internal processes, the action and movements of the 
infinitesimals of life? Think you not that such direct 
contact, such immediate and incontrovertible knowl- 
edge would be far superior to any advantage which 
his manufactured devices now bestow? It is even so. 



NEW PERCEPTIVE FACULTIES 357 

Thus will react upon man's perceptive apparatus 
the flood of light which the awakened intuition will 
shed upon them and thus will man rise higher, driven 
on by the current of life with the mass of materiality, 
to a point of complete spiritualization and take addi- 
tional steps in that direction which leads to Raja Yoga 
or the Royal Union with the divine life of the uni- 
verse. 

Before this step is taken, however, and before the 
passage from mechanics to biogenetics is made, as 
made it must be, man must win a complete mastery 
over matter. But this he will do ; for more and more 
he is learning to put all those forms of labor which 
are so exacting as to leave him no time for the 
development of his higher powers into the hands of 
machinery. He will not be free until he has done this 
well-nigh completely. This is the task of the intellect 
and with it man must win his way to these higher 
faculties which are destined to succeed the intellect 
whereupon he will be ushered out of a life bound and 
restricted by mechanics to a life of unimaginable free- 
dom, the intuitive life. 

The outcome of these new faculties of perception 
and the development of the intuition will be the spring- 
ing up of a new species of art that, turning away 
from appearances and sinking beneath or rising above, 
superficialities, will seek to portray in newly found 
colors, the plastic essence of things so that we shall 
have an art which pertains to the real, superseding 
that which pertains to the phenomenal. Language 
and the need of it will pass away; for man will have 
outgrown the use of signs and symbols in his com- 
munion with his fellows and will use the language of 



358 THE MYSTERY OF SPACE 

the intuition — direct and instantaneous cognition. 
Philosophy will be regenerated, re-created. Specula- 
tion will give way to truth and there shall be but one 
philosophy and that shall be the knowledge of the real. 
Mathematics, the royal insignia of the intellectual life, 
because it can deal only with immobilities, with seg- 
ments and parts and has no aptitude for the continu- 
ous flow, will yield its kingdom to a higher form of 
kinetics which will serve the intuitive faculty as mathe- 
matics now serve the intellect. Science will then be 
no longer empirical in its method; but a system of 
direct and incontrovertible truths. Religion will rise 
to meet these changes which will come in the path of 
human evolution; and faith will surrender its place to 
knowledge. Ethics, recast in a new mold, will deal 
with the new aspect of man's relation to his fellow- 
men. Man, for whose highest good these ultimate 
changes will come, will be a new creature, a higher 
and better man; and humanity shall evolve a new 
race. There shall, indeed, be "a new heaven and a 
new earth." 



THE END. 



BIBLIOGRAPHY 



American Mathematical Monthly: 

Vol. VIII, p. 161, P. Barbarin, for Le Mathematiche, 

Trans, by Geo. B. Halsted. See also pp. 31-35. 
Vol. VIII, 1 901, p. 217, Suppl. Report on Non-Euclidean 

Geometry, Geo. Bruce Halsted. 
Vol. IX, p. 59, Geo. B. Halsted. 
Idem, pp. 153-159. 
Vol. XI, pp. 85-86, Simon's Claim for Gauss. 

American Mathematical Society Bulletin, Lan- 
caster, Pa.: 

Vol. II, series 1895. 

Vol. XX, 1 91 4, pp. 409-412, R. C. Archibald. 
Annals of Mathematics, 2d series, 1902: 

Vol. XII, F. S. Wood. 
Astronomical Society Notices: 

Vol. XVI, p. 80, 1856. 

Ball's Short Account of the History of Mathematics. 
Boston Colloquium, for 1903, N. Y. 1905, Forms of Non- 
Euclidean Spaces, Vols. I-III, pp. 31, et seq., 189 1. 

Bulletin New York Mathematical Society: 
Vol. II, pp. 1-144. 
Vol. Ill, pp. 1, et seq., 1891, G. B. McClintock. 

Cambridge Mathematical Journal: 

Vol. IV, 1845, Cayley, on Analytical Geometry of 
n-Dimensions, 

Canadian Institute Transactions: 

Vol. VIII, pp. 315-339, 1909, August Kirschbaum. 

359 



360 BIBLIOGRAPHY 

CaylEY, A., vide Science, Vol. IX, pp. 59-63 : 
Amer. Math. Mo., Vol. II, pp. 102-106. 
Amer. Math. Mo., Vol. VI, pp. 59-65. 
Review of Reviews, June, 1895, pp. 693-694. 

Class Book of Non-Euclidean Geometry: 

Vide Science, XIII, pp. 462-465. 
Cornhill Magazine (n.s.): 

Vol. XXXVII, pp. 647-665, C H. Bryan, on the Fourth 

Dimension. 

Easy Non-Euclid: 

Monist, XIX, pp. 399-402. 
Educational Review: 

November, 1902, issue, pp. 346, et seq. 

Euclid's Parallel-Postulate, by John W. Withers. 

Flatland, by A. Square. 
Foundations of Geometry: 

Science, Vol. VI, pp. 487-491. 
Foundations of Mathematics, by Paul Carus. 
Fourth Dimension: 

By Herman Schubert, 
C. H. Hinton, 
Claude Bragdon. 
Fourth Dimension, Simply Explained: 

Edited by H. P. Manning. 
Fourth Dimension: 

Cornhill Magazine, n.s., Vol. XXXVII, pp. 647-665, by 
C. H. Bryan. 
Fourth Dimension, The Riddle of: 

H. A. Bruce, in Scientific American Supplement, Vol. 
LXVI, p. 146. 

Geometry, Foundations of, by O. Veblen : 

Popular Science Monthly, Vol. LXVIII, pp. 21, et seq. 

Geometry, Lobachevski's: 

Theory of Parallels, trans, by Geo. Bruce Halsted. 
Geometry, Non-Euclidean : 

American Mathematical Monthly, Vol. VII, pp. I23-I33* 



BIBLIOGRAPHY 361 

Geometry, Non-Euclidean, Report on Progress in: 

Reprinted from Science, n. s., Vol. X, No. 251, pp. 545, 
et seq., by Geo. Bruce Halsted. 

Geometry, Non-Euclidean, Thesis on: 
By M. L. Gerard. 

Geometry, Non-Euclidean : 

By Roberto Bonola, trans, by H. S. Carslaw, Uni- 
versity of Sydney, N.S.W. 

Geometry, Non-Euclidean : 

Julian Lowell Coolidge, pp. 46, et seq. 

Geometry, Non-Euclidean : 
By H. P. Manning. 

Geometry, Space and, in Light of Physiologic, Psychologic 
and Physical Inquiry, vide Open Court, 1906, by E. Mach. 

Halsted, George Bruce: 

Cayley, A., Biog. of, vide Amer. Math. Mo. t Vol. II, 

p. 102. 
Four-fold Space in Two-fold Time, vide Science, Vol. 

XIX, p. 319. 
Hoiiel, Biog. of, American Math. Mo., Vol. IX, pp. 

99-101. 
Lambert's Non-Euclidean Geometry, vide N. Y. Math. 

Soc. Bull., Vol. Ill, p. 79. 
Lobachevski, N., Biog. of, vide Amer. Math. Mo., Vol. II, 

pp. 137-139; Open Court, Vol. XIII, pp. 411-415. 
Message of Mathematics to Natural and Rational The- 
ology, vide Hibbert's Journal, Vol. VII, 1809. 
Non-Euclidean Geometry, Inevitable, vide Monist, Vol. 

IV, p. 483. 
Non-Euclidean Geometry, Historical and Expository, 
American Mathematical Monthly. 

Vol. I, pp. 70-72; 112-115; 149-152; 188-195; 
222-223; 259-260; 301-303; 345-346; 378- 
379; 421-423 (1894). 
Vol. II, pp. 10; 42-43; 67-69; 108-109; 144- 
145; 181-214; 256-257; 309-313; 346-348 
(i895). 
Vol. Ill, pp. 13-14; 35-36; 67-69; 109; 132- 
133 (1896). 



362 BIBLIOGRAPHY 

Vol. IV., pp. 10; 77-79; 101-102; 1 701 7 1 ; 200; 

247-249; 269-270; 307-308 (1897). 
Vol. V, pp. 1-2; 67-69; 127; 218; 290-291 

(1898). 
Review of Reviews, Vol. II, 1895, pp. 693-694. 
Space, New Ideas About, Popular Science Monthly, Vol. 

XI, p. 364. 
Helmholtz, H., On Origin and Meaning of Geometric 
Axioms: 

Vide Mind, Part I, No. 3, July, 1876. For replies and 
discussions of the above, vide Nature, Jevons, Vol. 
IV, p. 481; J. L. Tupper, Nature, Vol. V, p. 202; 
Helmholtz, in Academy, Vol. Ill, p. 52; Mind, 
Part II, April, 1878. 
Hermathena, Vol. XIII, pp. 491, et seq., 1906. 

Higher Space, A Primer of, by Claude Bragdon. 

Ingleby, C. M.: 

A Non-Mathematical Criticism of a Part of Professor 
Clifford's Lecture on "Curved Space," Macmillan's 
Magazine, October, 1872; also Nature, Vol. VII, 
1873. 

Journal of Mathematics, American: 

Vol. I, pp. 262, 276, 384, 386; Halsted (1878). 

Jour. Philos., Psychol, and Scientific Methods: 
Vol. VI, pp. 617, et seq. 

Kant, Emmanuel, trans, by Fred. Paulsen. 
Kant, Emmanuel, Prolegomena, trans, by J. P. Mahaffy 
and J. H. Bernard, pp. 35-39. 

Lambert, Biography of: 

American Math. Mo., Vol. II, pp. 209-211. 
Lie, Sophus: 

Science, Vol. IX, pp. 447-448. 
Lobachevski, N. : 

Open Court, Vol. XII, pp. 411-415, 

Amer. Math. Mo., Vol. II, pp. 137-139. 



BIBLIOGRAPHY 363 

London, Edinburgh and Dublin Philosophical Maga- 
zine: 

Vol. V, series 6, p. 559. 

Mathematics, Foundations of: Paul Carus. 
Mathematical Papers of Chicago Congress, p. 92. 
Mathematical Papers, pp. 55-71. 
Mathematics (a brochure) : 

By C. J. Keyser, Adrian Professor of Mathematics, Co- 
lumbia Univ. 
Mathematics, Higher: 

Edited by Mansfield Merriam and Robert S. Wood, 
1902, pp. 508, et seq. 

Mathematics, Spiritual Significance of: 

C. J. Keyser, reprinted from Columbia Univ. Quar. t 
December, 191 1. 
Memorial Volume of Scientific Papers: 

Pub. by Univ. of St. Andrews, pp. 3-45, D. M. Y. 

SOMERVILLE. 

Mind (Magazine) : 

Vol. I, 1876, pp. 303, et seq. J. N. P. Land. 
Vol. II, 1877, p. 40, J. N. P. Land. 
Vol. IX, pp. 522, et seq. 1900. 

Monist (Magazine): 

Vol. II, pp. 50, 321, Paul Carus. 

Vol. IV, 1894, Articles by G. B. Halsted and J. Del- 

boeuf. 
Vol. VI, 1895, Herman Schubert. 
Vol. VII, G. B. Halsted, on Saccheri. 
Vol. XIII, pp. 80-102; 218-234. 
Vol. XIV, Article by E. Mach. 
Vol. XV, p. 127, Paul Carus. 
Vol. XVI, 1906, p. 433, David P. Abbott. 
Idem, 1906, p. 65, C. J. Keyser, on Mathematical 

Emancipations. 

Nature (Magazine): 

Vol. VIII, pp. 8, 14-17, 36-37 (i873). 
Vol. XV, pp. 533-537 (1877). 
Vol XLV, P. 404(1891). 



364 BIBLIOGRAPHY 

Newcomb's Philosophy of Hyperspace: 

Science, Vol. VII, p. 212. 
New York Math. Soc. Bulletin: 

Vol. Ill, p. 79 (1893), F. S. Wood on Forms of Non- 
Euclidean Space. 

Open Court (Magazine): 
Vol. XVI, pp. 513-521. 
Vide December, 191 3 edition, p. 757. 

Philosophy of Mathematics, by Paul Carus. 
Philosophical Review: 

Vol. V, 1896, p. 175, F. C. S. Schiller. 

Idem, p. 357, J. H. Hyslop. 

Vol. VII, p. 615. 

Vol. X, pp. 113, 229, 314, 375, 488, 579, 583. 

Vol. XII, p. 493- 
Popular Science Monthly: 

No. 66 (1905), PP. 639-646. 

Vol. LXVII, pp. 639, et seq. 

Vol. LXVIII, pp. 21, et seq. 

Vol. LXXVIII. p. 554 (191 1 ), Samuel Moffat 
Weyer. 
Popular Astronomy: 

Vols. VII and VIII (1900), G. B. Halsted. 

Vol. XVIII, pp. 42, et seq. (1910, W. H. Pickering. 

Vol. LXXV, p. 179, W. W. Payne, on Attraction and 
Figure of the Earth. 

Vide No. 84 (1901), pp. 187-190. 
Popularization, The, of Non-Euclidean Geometpy: 

Amer. Math. Mo., Vol. VIII, pp. 31-35. 
Psychological Bulletin: 

Vol. VIII, p. 22 (191 1 ). 



Queen's Quarterly: 

Vol. XX, pp. 431-446, A. L. Clark (19 13). 

Revue Generale des Sciences: 

No. 23, trans. Nature, Vol. XLV, 1892. 



BIBLIOGRAPHY 365 

Royal Astronomical Society Proceedings: 

Vol. XVI, pp. 80-85. 
Royal Society Proceedings, London: 

Obituary Notices of Deceased Fellows, Vol. XVI, p. 69 
(1867-68). 
Royal Society of N. S. W. Journal and Proceedings : 

Vol. XXXV, p. 243 (1902), G. H. Knibbs. 

Science (Magazine): 

Vol. Ill (new series), No. 58, 1896. 

Vol. VI, pp. 487-491 (1897)- 

Vol. VII, pp. 1-7, Philosophy of Hyperspace, S. New- 

COMB. 

Idem, p. 861 (1898). 

Vol. XIII, pp. 462-465. 

Vol. XIV (new series), pp. 711-712, G. B. Halsted. 

Vol. XIX, pp. 401-413. 

Vol. XX, pp. 353-456, Josiah Royce. 

Vol. XXII, pp. 270-271. 

Vol. XXXV, No. 906 (1912). 

Vol. XXXVII, pp. 885, et seq., C. J. Keyser, on Con- 
cerning the Figure of the Universe of Space, 
Science et L'Hypothese, R. Poincare. 
Scientific American Supplement: 

Vol. LXVI, pp. 14-16 (1908). 
Scientific Romances, C. H. Hinton. 

Thought, New Era of, C. H.* Hinton. 

Westminster Review: 

Vol. CLII, pp. 675, et seq. 

Zollner, J. C. F., On Space of Four Dimensions: 

Quar. Jour, of Science, Vol. VIII (1878), pp. 227-237, 



INDEX 



Abeyance, higher faculties held in, 
until lower are fully developed, 

334 

— intuition kept in, 338, Jf 
Ability, to sense the inaudible and 

the invisible, 342 
Absolute, no room for the, 101 
Abstract thought, invigorating 

power of, 33, 40 
Abstractions, realizing, 144, 294 
Absurdity, analysis inevitably ends 

in an, 319-20 
Acrobatics, mathetic, 146 
Acromegaly, 347 
Action, automatic, of the intellect, 

253-4 

— engendering, of life on the form, 

33o 
Activities, space, as physical and 

chemical phenomena, 229-30 
Actualities of the physical world 
pushed over into the concep- 
tual, 119 
Actuality imperceptible to the in- 
tellect, 126 
Adaptability of life, 330 

mathematical laws, 37-39 

Adaptation, purpose of, 327-8 
Adaptations made by an organism, 

162 
Additional freedom bestowed by 

higher faculties, 334 
Affinity, zones of, 22, 124 

367 



Agency, interpretative, intellect as 

an, 166-7 
Agent, ego as an, of the Thinker, 

243,/ 

— engendering, life as an, 331 

— of creation, life as an, 343 
Ahmes, an Egyptian priest, 44 
Alchemy of psychogenesis, 225 
Alcohols, eight different, from one 

formula 155 

Algebraic quantities and space, 
125-7 

Allowability of the rules of logic, 163 

Aloofness, the, of scientific dogma- 
tism, 340 

Alphabet of space-genesis, 237 

— the geometric, 193 
Amenability of mind to the laws of 

evolution, 28-31 
Analogies, use of, to popularize the 

four-space, 128-9 
Analogy, difference between two 

processes illustrated by, 74 
Analysis incapable of dealing with 

life, 319 

— the manifold, the fiat of, 77 
Analyst and the manufacture of the 

space-manifold, 77 

— disregards the conformity of the 
nature of things, 84 

Analytics and the mechanical origin 
of the universe, 40-1 

— perisophical nature of, 320, f 

— the four-space, a curiosity of, 40 



368 



INDEX 



Anchorage, fourth dimension denied, 

265 
Angularity of consciousness, 120 
Answer of the senses to new needs, 

332 

— the, of consciousness to realism, 

197 

Anti-Euclidean geometry, 69 

Apotheosis of the definition, 78 

Apparatus, man's perceptive, 357 

Apparent vassalage of life, 329-30 

Appearance, dynamic, space as, 
104-7 

Appetences, materialistic, of empiri- 
cists, 340 

Appulses of life against matter, 
328 

Appurtenances of materiality, 331 

A priori, 185, Jf 

knowledge, 110-5 

Aprioriness, 112, 116 

Apriority, the principle of, 113, 223 

Arbitrariness of the common men- 
surative quantities, 41 

Archeological evidences, 332 

Archimedes, 70 

Aristotle, 322 

Arrangement of the contents of the 
kosmos, 317-8 

Art and the criterion of truth, 323 

higher consciousness, 181 

— evidences of new faculties in the 

field of, 335 

Assemblage of mathetic contriv- 
ances, 335 

Assumptions, category of, 189 

— procedure based upon, disadvan- 

tages of, 163 
At-one-ment, 224, 252, 270 

— defined as the end of evolutionary 

activity, 1 
Attachment of the consciousness to 
the order of becoming, 305 



Attainment, the difficulty of, by the 
intellectual method, 206 

— of space-consciousness, 225 
Attitude, the need of a more liberal, 

34o 

— towards things spatial, 306 
Audital nerves, notes that tran- 
scend the grasp of, 355 

Aura, pulsating, 348 

Automatism of the intellect, 253, ff; 

338 
Avenues of knowledge opened to 
inner perceptive faculties, 353 
Awakening of the faculty of aware- 
ness in a new domain, 90 
pituitary body and the pin- 
eal gland, 353 

— pituitarial, general results of, 

355-8 
Awareness, a determinant of concep- 
tion, 120 

— as a gauge of the existence of 

things, 161-202 

— degrees of, 165 

— hyperspace, a symbol of a more 

extensive, 180 

— progress in, 171 

— Thinker's sphere of, 274, 283 



Baltzer, 66 

Barrier, the, to the Thinker's certi- 
tude, 186-8 

Barriers, freedom determined by 
absence of, 280 

Becoming, endless labyrinth of, 220 

— the kosmos, in a state of, 265 

— the order of, 305 
Becomings, infinity of, 234 
Becquerel rays/ 336 

Being, identification of consciousness 
with, 205 

— kstho '.otic, 210 



INDEX 



369 



Being, the interiority of, 290 

— the numericity of, 309 

— the range of, coextensive with 

reality, 168 
Beings, discarnate, 154 
Beltrami, Eugenio, 1-2, 6$-$, 66-7 

and the manifold, 317 

and the pseudosphere, 64-5 

Besiegement of matter by life, 329- 

30 
Bessel,on geometry as incomplete, 58 
Bewilderment, new realities cause, 

169-70 

— of the mind, 182 
Biogenetics, passage from mechanics 

to, 357 

— turning to, from mechanics, 322 
Birth, as a fourth dimensional proc- 
ess, 159 

Blind men and the elephant, 257 
Bolyai, Janos, 2-3, 46, 60, 66, 83, 87 
Boundaries of a hypercube, 136-8 
Brain, the physics of, 292 
Brain-consciousness, 184, 186 

— standards of, 190 
Brotherhood, and the tendencies of 

the Thinker's consciousness, 
301 

— the keynote of the intuition, 252 
Bulwarks, the formal, of geometry, 

77-9 



Carpet, the sensible world as a, 196 
Carus, Paul, on four-dimensional 
space, 254 

on metageometricians, 102 

on space, no 

on the representation of the 

tesseract, 136-7 
Cayley, 3-4, 54-5, 66-7 
Cell-activity as a performance a 
priori, 114 



Cell-colony, 250, 301 
Cell-consciousness as an aposteri- 

oristic phenomenon, 185 
Centers, pyknotic, 232 
Certainty, geometric, the basis of, 

77-8 

— of mathetic conclusions, provi- 

sional, 37 
Chaogenic period of involution, 

208,/ 
Chaogeny, 231, 233 

— the evolution of chaos into 

order, 4 

— the laboratory of, 193 
Chaomorphogeny, defined, 4, 231-2 
Chaos, as egg-plasm, 265 

— duration of, 268 

— movement of life in, 307 

— shores of, 307 
Chaos-geometricity, 307 
intellectuality, 307 

— -materiality, 307 

— -spatiality, 307 

— -Theos-Kosmos, 215, 233-4, 237 
a triglyph symbolizing kath- 

ekos, 1 1-2 
Character of the universe, fixed by 

consciousness, 162 
Characteristics, changes in the, of 

reality, 169 

— fourth dimensional, of the ether, 

157 

— minds of similar, fall into zones of 

affinity, 124 

— of non-Euclidean space, 72-3 
Chasm, kosmic, between the real and 

the ideal, 107 

Chemists, speculative, and the 
fourth dimension, 155 

Chrism, sacred, of creative men- 
tality, 91 

Circle, significance of process of 
squaring, 109 



37° 



INDEX 



Circle, the symbology of, 309 

Circuits, closed, our interests as, 167 

Clairaudience, 352 

Clairvoyance, 352 

Clairvoyant perception, 349 

Clause, enabling, of metageometry, 
76-7 

Clavius, Christoph, 46, 52, 83 

Clifford, 66 

Code, psychic, for systematizing cog- 
nitions, 190-1 

Cognition, instantaneous, organ of, 
258 

— intimacies of direct, 205 

— the method of, 186-8 
Cognitions, intuitive, 145, 192-3 

— psychic code for systematizing, 

190-1 
Commensurable quality, dimension 

as, 140 
Communal consciousness, 270, Jf 
Communalization and the intuition, 

301 
Conarium, the, 345 
Conceivability and evolution of 

mind, 24 

— ultimate range of, 278 
Conceiving, power of, derived from 

sense-experience, 26 
Concept, as a shadow, 119 

— the hyperspace, gradual rise of, 

27-8 
Conception and awareness, 120 

— every, based upon prior experi- 

ence, 25 

— freedom, the, of the mind from, 

28 
Conceptions, symbolic, dependence 

placed upon, 147 
Concepts, and the Thinker, 244 

— intuitograms as, 248 

— quality of, dependent upon sense- 

experience, 26 



Concepts, the perception of, 255 
Conceptual, related to the objective, 
291 

— the, and mathematicians, 71 
Conceptualization, the act of, 4 

— the power of, dependent upon 

evolution of mind, 25 

Conclusions, mathetic, complex de- 
pendence of, 37 

Concrete hewn into shape by the 
intellect, 295 

Congruency, the, between intellect 
and the universe, 319,$ 

Congruity, the, between concepts 
and objects, 120 

Connection, the, traced out between 
reality and object, 126 

Co-originality of things, 303 

Consciousness and kosmos, graph 
representing, 271 

the character of the sensible 

world, 166 

wake of life, 205 

time, 224 

— angularity of, 120 

— a priori, 185 

— a replica of, and judgments, 205 

— as a scale, 165 

barrier to ultimate knowledge, 

207 

determinant of dimension, 145 

life, 295 

variable quantity, 171 

— attachment of, to the order of 

becoming, 305 

— dawn of, and dimension, 179 

— degrees of, 162 

— deprived of the validity of no- 

tions, 314 

— evolution of, 250 

— expansion of, 164, 272 

— extended, 89-90 

— focus of, 290 



INDEX 



37i 



Consciousness and the planes of the 
kosmos, 163 

— fragmentary view of, 260 

— freedom of, in the genus homo, 332 

— habitation of, 278 

— higher, and art, 181 
planes of, 178 

— history of, 298-9 

— identification of, with objects, 

189-90 

— intuitional, 255 

— keyed to the entire range of 

reality, 167-8 

— kinds of, 270, f 

— kosmic, Elysian fields of, 235 

— of the primitive man, 253 

— organ of, 184,/; 90 

— psychics of, 292 

— raised from the sensuous to the 

conceptual, 72 

— recipient of truths from dual 

sources, 26 

— sensible world as instrument of, 

199 

— triple presentation of notions of 

space to, 72 

— turned inward, 283 

— unification of the states of, 192 

— youthful, of mankind, 122 
Consequences, the science of, 85 
Considerations, mathematical, and 

reality, 128 
Consistency, kosmic, 1 74 

— self, of hyperspatial hypotheses, 

67 

— the criterion of geometry, 86 
inconsistence of, 1 73 

metageometrician curtailed by, 

85 
Construction, idealized, cannot be 
objectified, 144, 276 

— ideal, the meaning of, 4 

— mental, trafficking in, 154 



Consubstantiality of intellectuality 
and spatiality, 331, Jf 

Continuity of the psychic plasm, 
260 

Contrivances, mathetic, the passing 
of, 241 

Convergence of parallel lines, 315 

Convolutions, duplex, 146 

Coordinates, as spatial determi- 
nants, 203 

— of hyperspace, 94, 97, 99, 115-6 

— systems of, 301, Jf 

— three, 132, 232 

Copolarity of ideas and objects, 127 
Corpora quadrigemina, 345 
Corpuscular orbits of particles, 152 
Cosmos, significance of, 4 
Creation, impregnated screen of, 232 
Creations, the, of the intellect, 173 
Criterion of geometry as consistency 
and convenience, 86 

truth, 323-4 

Crookes, 344 

Cube, as succession of planes, 148 

— generation of the, 144 

— illustrated, 133 

— the generating, 134 
Curiosities, analytic, energy spent 

in elaborating misappropri- 
ated, 39 

Current, electric, as component in 
the fourth dimension, 153 

Curvature of space, the doctrine of, 
formulated by Riemann, 5 

valid element in the notion 

of, 305 

— the measure of, 61 

Curved space and metageometri- 
cians, 316 



D'Alembert, note, 51 
Dante, 181 



372 



INDEX 



Darwin, 181 

Day, the Great Kosmic, 213 

De Tilly, 66 

Death, as fourth dimensional unity, 

159 

— of the intellect, 195 
Decrement of the diametrical ratio 

to the circumference of circle, 
312 
Deeps, fearful, of kosmic mind, 71, 

9i 

Definition, apotheosis of, 77-8 

— as an arbitrary determination, 

100 

— the deification of, 37 
Degrees of realism, 164-5 
Deiform, basic idea of the, 5 
Demarkation (between reality and 

phantasy, the line of, 173 

Deposits of life, 173 

Descartes, 322 

Designs, cut out in materiality by 
life, 264 

Details, the power to dispose of in- 
numerable, 303 

Determinations, geometrical, the 
necessity of, inheres in logical 
deductions, 78 

— the factors of conscious, 162 
Determinative period of mental 

development, 31 

Development of the intuitive fac- 
ulty, future, 188 

Diacritics of life, 286 

Diameter, the ratio of, to the cir- 
cumference, 3 1 1-2 

Difference between concept and 
thing, 1 2 1-2 

— between mathematical and per- 

ceptual space, 120 

— between the ideal and the actual 

is dynamic, 108 
Differential, among minds, 124 



Difficulty of imagining the fourth 
dimension, 90 

— of propagating an intuition, 315 
Difficulties in the acquisition of real 

knowledge, 325 

— of hyperspace, the logical, 139- 

41 
Dilemma of metageometricians, 97-8 
Dimension, a distinct stage in psy- 
chogenesis, 27 

system of space measurement, 

5 

— and analysis, 163 
D'Alembert, note, 51 

the action of a tartrate in, 

156 

— as an arbitrary contrivance, 262 

assemblage of elements, 93 

direction, 140 

extent, 98 

lying near the surface of things, 

159-60 
space, 96 

— current definitions of, 98-100 

— denied legitimate anchorage, 265 

— determined by consciousness, 145 

— does not explain spiritualism, 155 

— evolution of the fourth, 44 

— fourth, direction of, 134 
as "jack of all trades," 156 

— impossible to actualize, 124 

— key, to non-Euclidean geometry, 

90 

— movement of a plane into, 147-8 

— no motion of material masses into, 

125 

— not a priori, 116 

— proof of the existence of the 

fourth, 136 
Dimensionality/ and the intellect, 
200 

— as dependent upon the will of the 

investigator, 95 



INDEX 



373 



Dimensionality, conception of, 
purely conventional, 92 

— logical difficulties which beset, 97 

— of perceptual space, 72 

space, the four, 142-3 

Dimensions, four, no basis in con- 
sciousness for, 172 

— piled upon one another, 300 

— vanity, the, of segmenting space 

into many, 299 
Direction as dimension, 99 
Disappointment with the sensible 

world, 1 20-1 
Discoveries, metageometrical, as 

excrescences of mental unfold- 

ment, 131 

— never the result of methodic re- 

flection, 29-30 
Disorder, the edict of, and space, 

230 
Disorderliness, the fringe of, 277 
Dispossessal of the intellect by the 

intuition, 195 
Dissimilarities, besetment of, 291 
Distinctions, between conceptual 
and perceptual space, funda- 
mental, 73-5 
Diversity, a result of the fragmen- 
tative tendency of life, 6 

— as transfinite quantity, 42 
Divinity, prize of, won by the intel- 
lect, 43 

Duadic phase of evolution, 210-1 
Duodim, denned, 6; 128, 129, 134, 
177 

— consciousness, 163 
Duopyknon, 212, 213; 

— the meaning of, 6 
Duopyknosis, 208-10, 218 

— as a stage in the evolution of 

space, 6 
Duration and space, 224 

— eternal, 234 



Dynamic appearance, space as, 104, 
107 

Dynamism, of the intellect, func- 
tional, 303 

Earth, the nebulous youth of, 335 

— a new, 358 

East effloresces as West, 306 
Ego, the, 270 

— and percepts, 244 

— as sovereign, 244 

— Thinker's dependence upon, 245 

treatment of, 246 

Egopsyche, 265 

— as an agency of self-conscious- 

ness, 258 

— defined, 7 

Egos, compared to choppy sea, 

256-7 
Egypt, birthplace of geometry, 44 



E 

Eisenlohr, 44 

Elaborative period of the non- 
Euclidean geometry, 66-70 

of mental development, 31-32 

Element of evolution, basal, 259 
Elements of the non-Euclidean 

geometry, 79-80 
Elephant and blind men, 257 
Ellis, Wilmot E., on ether as four- 
dimensional, 157-9 
Embrace of direct cognition, 205 
Engenderment of space, 260 
Ensoulment of the world-plasm, 329 
Entities, hyperspace, and the phe- 
nomenal world, 128-130 
Enumeration, reformation of the 

system of, 38 
Enveilment of consciousness, 273-4 
Environment, artificial and natural, 

162 
Epiphysis, the, 345 



374 



INDEX 



Equidistantial, described by hyper- 
spatial perpendicular, 80 
Essence, elemental, as world-plasm, 

329 

Ether as possessing fourth dimen- 
sional characteristics, 157 

Ethics recast in a new mold, 358 

Euclid, 46, 70, 83, 263 

— and the parallel-postulate, 45 

— never-dying elements of, 53 
Evidences, mathematical, exempli- 
fies intellectual evolution, x 

— the vanity of fragmentary, 204 
Evolution, a continuous process, 327 

— and the norms of reality, 175 

— basal element of, 259 

— commencement of, 232-3 

— intellectual, forward movement 

of, 184 

— kosmic, vicissitudes of, 215 

— laws of, govern mind, 28 

— mental, results of, 122 

— of faculties, synchronous with 

evolution of matter, 332 

material characteristics of the 

earth, 335 
organs, time required for, 188 

— preparation of the field of, 218 
Exaltation of matter into spirit, 

329 

Examples of new perceptive evi- 
dences, 341-3 

Existence on a higher plane, states 
of, 162 

Experience, corroborative testimony 
of, denied hyperspace, 263 

— prior, and conception, 25 
Experiences, spatial, systematiza- 

tion of, 78 
Extension, space as an unbounded, 

61 
Extra-spatiality, degraded into spa- 

tiality, 261 



Extravaganza, mathematical, and 

the fourth dimension, 156 
Eyes, as Roentgen rays, 353 

F 

Fact-mass, 289 

Factors of conscious determinations, 
162 

— four, of the criterion of truth, 

324-5 
Facts as facets of truth, 284 

— attempts to reform, 304 

— logic as symbolism of, 287 

the modeler of, 288 

Faculties, dual derivation of, 162 

— evolution of, synchronous with 

evolution of material qualities, 
332,/ 

— extended, 239 

— foreshadowed by the hyperdi- 

mensional, 131 

— higher, man must win his way to, 

357 

— new faculties evidenced by four- 

space, 24-29 

— rudimentary nature of, 341, jf 

— the source of, 349 

Faculty and the intellect, 247, jj 

— as transmissible character, 251 

— determined by function, 336 

— greatly extended, 117 

— higher than the intellect, 126 

— I-making, self-consciousness as, 

243 

— intuitive, 185 

— of perception in higher worlds, 353 
awareness, the awakening of, 

89-91 

— overshadowed by the intellect, 

188 

— of perception in higher worlds, 

353 

— rudimentary condition of, 192, / 



INDEX 



375 



Faculty, outcropping of, 249-51 

Failure of efforts to justify the ob- 
jective existence of four-space, 
125-6 

Failures at solving the parallel-pos- 
tulate, outcome of, 48, 83 

Faith, dispossessed by knowledge, 
358 

Fay, mathetic, 160 

Fechner, 39 

Finity and unboundedness of space, 
76 

Flammarion, Camille, 341-2, 344 

Flexity, as property of hyperspace, 

63 

Flexure, space as a, 305 

Fluxional, between sense objects and 
ideal representations, 122-3 

Fluxion, psychic, as difference be- 
tween memory image and 
object, 7-8; 122 

Focus of consciousness, 163, 290 

the intellect, 310 

Fohat, and the creation of morphons, 
219 

— as creative energy, 213 

— Creator, 8 
Fohatic energy, 226 
Form, as vehicle of life, 330 

— definition of, 8 

— driven back over the path of its 

genesis, 327 

— pure, cannot exist in nature, 

294 

— the idealty of, no 

— the universe not a pure, 108 
Formative period of mental devel- 
opment, 31 

of non-Euclidean geometry, 55 

Formula, eight different alcohols 

from one, 155 
Four-dimensionality, justification of, 

176 



Four-space, a curiosity of analytics, 
40 

— and Riemann, 27 

— as a divertisement, 175 

— consciousness does not act in the, 

172 

— existence of, denied, 17 1-2 

— movement of matter in, 157 

— reality of, glibly proclaimed, 154 

— the, 8, 240 

denizen c of 353 

domain of, 154 

study of, 124 

— use of analogies to popularize, 

128 
Fourth dimension, analogical rea- 
soning of, 177-9 

— as a transcendental problem, 140 
an attitude of the intellect, 200 

— electric current as a component 

in, 153 

— imaginability of, 106-7 

— Simon Newcomb on, 125 
Fragmentariness of the intellectual 

method, 164 
Fragmentary view of the universe, 

260 
Fragmentation, tendency to, 296 

— harks back to cell-division, 301 
Freedom, a new, 338 

— determined by absence of bonds, 

280 

— mathematical, 60 

— mental, 66-7 

— now dawning for the mind, 32 

— of consciousness in genus homo, 

332 

— regal freedom of the mind, 118 

— three degrees of, 125 

— unrealizable, for the Thinker, 255 
Fringe, chaogenetic, 308, 309 

— of disorderliness, 257 
kathekosity, 229 



376 



INDEX 



Frischauf, 66 

Full, the universe as a, 310 
Function determines faculty, 336 
Functions back of latent faculties, 
indefinite, 343 

— cellular and histologic, 253 
Functioning, instinctive, of the in- 
tellect, 289 

Fundamentals, totality of, kosmic, 
265 

G 
Gamut of realism, 169 
Gauge, awareness as a, 161-202 
Gauss, Charles Frederick, 8-9, 56-8, 

59, 174 
as formulator of the non- 
Euclidean geometry, 57 
Geminos of Rhodes, 45, 52 
Generability, as property of space, 62 

— of hyperspace, Keyser on, 144 

space by lines, 143 

Generation, the, of the hypercube, 

134 

— of the hypertetrahedron, 135-6 
Genesis, of space, 211, 227, Jf 
the earth, its nebulous youth, 

335 

form, 327 

sensible world, 167 

Genus homo, freedom of conscious- 
ness in, 332 

Geometricity, 266 

Geometries, non-Euclidean, based 
upon a negation of the latent 
geometrism, 262-3 

— three possible, 54 
Geometrism, engenderment of, 262 

— established by life, 264 

— kosmic geometrism, establish- 

ment of, 231 

— latent, 257 

— native geometrism of space, 262 

— rediscovered by the intellect, 9-10 



Geometrism, the basis of, 237, 261 
Geometry, anti-Euclidean, 69 

— and the study of magnitudes in 

space, 203 

— artificial, 262-4 

— a two-fold, 59 

— breakdown of, 266 

— determinative period of the non- 

Euclidean, 61-4 

— diverse systems of, 323 

— Euclid's Elements of, 53-4 
I — formal bulwarks of, 77-9 

— Imaginary Geometry, The, 60 

— natural geometry, 261 

— Plato and the divine geometry, 

193 

— possible systems of, 174 

— radical essence of pure, 85 
Geometry, non-Euclidean, at vari- 
ance with the parallel-postu- 
late, 83-5 

based upon a misconception, 

91 
determined by qualitative dif- 
ferences, 140 

elaborative period of, 66-70 

first published treatise on, 60 

formative period of, 55 

the final issue of, 88-9 

growth and development of, 92 

invalidation of, 174 

key to, 90 

Lambert's Non-Euclidean ge- 
ometry, note, 51 

popularization of, 66-7 

Schweikart's treatise on, 59 

self -consistence of, 57 

some elements of, 79-80 

superperceptual knowledge of, 

72 
Geometry, non-Legendrean, 70 

— symbolic, and commensurable 

quality, 145 



INDEX 



377 



Gerling, as correspondent of Gauss, 

58 
Germ-plasm, continuity of, 260 
Giantism, 347 
Glimpse, a, of the reality of space, 

314 

Glorification of the flesh, 227 

Groups, transformation, discovery 
of, 30 

Guide-posts to a new domain, 273, 
277 

Gulf, interposed between manifesta- 
tion and non-manifestation, 
206 



Halley, memoir of, mastered by La 
Grange, 50 

Halstead, G. B., note, 49, 66; note, 79 

Heindel, Max, note, 353 

Helmholtz, 39, 66 

Hewer, the, of the concrete, the in- 
tellect as, 295 

Hinton, C. H., 154 

— on the fourth dimension, 153 
History, the, of mind, three great 

epochs of, 31-32 
Homogeneity, the, of realism, 165 
Hoiiel, J., 60, 66 
Hudson, 344 
Humanity and the exalting power of 

life, 331,/ 
Hypercube, boundaries of, and 

mirrors, 136-9 

— the generation of, 134, 145 
Hyperdimensional, as a prophecy of 

new faculties, 131 
Hyperspace, 175 

— a monument to the intellect, 356 

— and the involved procedure of 

arriving at a recognition of its 
relations, 73 

— and the passage thither, 293, Jf 



Hyperspace as a figurative moun- 
tain-peak, 200 

movement, 44-68 

an all-powerful something, 160 

idealized construction, 10 

evidence of new faculties, 24 

ignes fatuii, 154 

illusion, 202 

— a symbol of higher conscious- 

ness, 180-4 

— concept of, as an evolutionary 

quantity, 44-5 

— confounded with real space, 91 

— creation of, 335 

— denied the corroborative testi- 

mony of experience, 263 

— direction of, baffling to mathe- 

maticians, 146 

— discovery of, a sign of mental 

evolution, 180 

— domain of, a fairyland, 239 

— magnitudes of, the non-sensuous, 

72 

— mysterious hiding place in, 129 

— the logical difficulties of, 139- 

141 

— the six pillars of, 62-3 
Hyperspatiality as the toys of child- 
hood, 241 

Hypertetrahedron, the, 135 
Hypertrophy of the pituitary body, 

347 
Hypervolume, the, 134 
Hypothesis, a superfoetated, 141 

— four-space, utility of, 129; 154 
Hypotheses, admissibility of, 118 

— incompatibility of the non-Eu- 

clidean, 67; 71 

— Riemannian, 80 

— solution of, Bolyai, 31 

Hyslop, James H., on the logical 
difficulties of hyperspace, 139- 
141 



378 



INDEX 



Icosahedron, examination of the, 

285 
Ideal, and the real, kosmic chasm 

between, 107 

— perceptual value of, 275, Jf 
Ideas and words as symbols, 126-7 

— Malebranche on, 222 

— realism of, 24-35 

— the symbolism of, 205 
Identification of consciousness with 

being, 205-6 

— with the objects of study, 189-90 
Identity of things-in-themselves, 353 
Ignes fahiii and hyperspaces, 154 
Illustration of plane-rotation, 148- 

150 

the tesseract, 133 

Images, totality of, recoils upon us, 

167 

Imaginability of the fourth dimen- 
sion, 90, 106-7 

Imagination, premises of, the math- 
ematical, 146 

Impossibility of plane-rotation, the 
structural, 15 1-3 

Impressions, the symbolism of, neu- 
rography, 186 

Impulse, the satisfaction of the orig- 
inal creative, 310-1 

Incomprehensibility of reality to 
the intellect, 126 

Incongruity, life estranged by a rad- 
ical, 325 

Individual as space, 223 

Ineptitude of intellectual determin- 
ations for vitality, 314 

Infinite, interpreted in the terms of 
the finite, 82 

Infinitesimals of unity, numbers as, 

41 
Infinity as a process, 109 

— of becomings, 234 



Infinity of parallels through a 

given point, 70 
space, a capital illusion, 195 

— the concept of, 277 

innate dread of, 103 

relativity of, 194 

Influence of abstract thought, 33 
Kant on the non-Euclidean 

geometry, 49-50 

La Grange, 51-2 

the intellect, 315 

Infundibulum, the, 345 

Inner organs of perception, 352 

Innermost, the, realities of things, 

356 
Insouciance of the geometer, 96, 294 
Instant-exposure and intellect, 311 
Instrument, intellect likened to a 

color-bearing, 313 

— for the measurement of the pas- 

sage of space, 297 

— of consciousness, the sensible 

world as, 199 

life, form as an, 328 

Integers, as fractional parts of unity, 

41 
Intellect and its domination of the 

intuition, 333 
final union with the space- 
mind, 194 

topography, 312 

spatiality, 263 

the deposits of life, 173 

designs cut by life in ma- 
teriality, 264 

dictum of Sensationalists, 

26 

instant-exposure, 311 

— intuitive faculty, 247, jj 

prize of divinity, 43 

— as a color-bearing instrument, 

313-4 
searchlight, 168 



INDEX 



379 



Intellect as a fashioner of phenom- 
ena, 199,/ 

hewer of the concrete, 295 

sole interpretative agency, 

166-7 

— automatism of, 253, 338 

— cannot seize life, 282 

— crowned by a diadem of gold, 338 

— dominated by the intuition, 250 

— fashioned for matter only, 231 

— follows in the grooves of logic, 

294 

— hyperspace as a monument to, 356 

— in the field of vitality, ix 

— its aptitude for starts and stops, 

292,302 

instinctive tendency to frag- 

mentate, 296 

— quality, determined by mathe- 

matics. 337 

— makes for individuality, 252 

— misses the ceaselessness of life, 201 

— modus vivendi of, its influence 

upon knowledge, 184 

— the constitution of, 291 

cut and mode of, 167 

focus of, 310 

illusion of, 246 

illusionary character of, 323 

incomprehensibility of reality 

to, 126 

instinctive functioning of, 289 

judgments of, 302 

moods of, 202 

predominating tendency in, 

320-1 

scientific tendence of, 165-6 

struggle of, against dispos- 

sessal, 195 

— unsuited to deal with reality, 

322,/ 
Intellectuality and reality, 304 
spatiality, consubstantial, 331 



Intellectuality as coextensive with 
spatiality and materiality, 236 

— the source of, 260 

Thinker makes his own, 242 

Intelligence and automatism, 253, jj 

— the Thinker as a pure, 243 

— transfinite intelligence and the 

degrees of realism, 164 
Intent, the kosmic, of the intellect, 

326 
Interests, the sphere of our, as 

closed circuits, 167 
Interior, the great, 290 
Interiority of being, 290 
Interpretation, the standards of, 

vary as consciousness varies, 

171 
Interstices of materiality, 264 
Intuition and brotherhood, 252 

communalization, 301 

the riddle of spatiality, 325 

— as dispossessor of the intellect, 195 

— cannot be served by mathematics, 

337 

— dominated by the intellect, 333 

— held in abeyance, 338, f 

— its domination of the intellect, 

250 

— the development of a spatial, 

not absurd, 145 

need of a sympathetic atti- 
tude towards, 249 

results of the development of, 

357 
Intuitional consciousness, 255 

— the superiority of, over the ra- 

tional, 187 
Intuitions and the lead of life, 191 
Thinker, 2 7 

— free, mobile and formless, 166 

— the conceptualization of, 248 
humility of, 165 

nature of, 185 



3 8o 



INDEX 



Intuitograms, as concepts, 248 
Intuitograph as means of contacting 

the egopsychic consciousness 

by the Thinker, 10 
L — as superconcepts, 255 

— the difficulty of transmitting, 

3i5 

Invariability, the vaunted, of the 
laws of mathesis, 37 

Invariants, psychological, 24 

Investigations, metageometrical, and 
the new sense-organs, 354 

Involution, as antithesis of evolu- 
tion, 10-11 

— kosmic involution, 226 

— of matter, 328 

— the movement of, 219 

seven stages of, 208-1 2 

Ions, creation of, 219; 

— magnitude of a hydrogen ion, 225 



Judgments a priori, Kant on, 85-6 

— and the faculty a priori, 190-1 
zones of affinity, 1 24 

— based upon a replica of conscious- 
ness, 205 

— no trustworthy, can be predicated 

upon fragmentary knowledge, 
282 

— of the intellect, 302 

— the lessening of error in, 256 
more complex the more at 

variance with the nature of 
things, 75 
synthesis of, 257-8 

— valid judgments long delayed, 

1 70-1 
Judicative power of mathematics, 

180 
Justification for a multi-dimensional 

quality in space, 26 2 

— of four-dimensionality, 176 



Justification of sense-deliveries by 
ODe another, 76 

the existence of the fourth di- 
mension, 125 



Kant, 85, 181, 182, 322 

— and the faculty of thinking, 261 
idea of space, 322-3 

— influence of, on the hyperspace 

movement, 49-50 

— on the nature of things, 119 

space as an intuition, 115 

Kathekos, 233-4 

— as chaos, 266 

symbol of Chaos-Theos-Kos- 

mos, n-12 

— symbology of, 234 
Kathekosis, note, 227 
Kathekosity, fringe of, 229 

— rock-bound coast of, 267 

— significance of, 12 
Kathekotic consciousness, 272 

— period, 209 

Key to the mysteries of nature, the 

fourth dimension as, 131 
Keyser, Cassius Jackson, and free- 
dom of the mind, $$ 

on attitude of metageo- 

metricians, 71 

dimensionality, 94-5 

f our-dimensionality of 

space, 142-3 
generability of hyper- 
space, 144 
Klein, Felix, 12, 54-5, 66-7 
Knowledge, all, relative, 101 

— barrier to the certitude of the 

Thinker's, 186-7 

— fabric of, 196 

— hypothetical nature of, 189, f 

— immeasurable realm of, laid bare 

by the telescope, 298 



INDEX 



38i 



Knowledge, mathematical, apriority 
of, questioned, 37 

— nature of the non-Euclidean, 

superperceptual, 72 

— real, difficulties of acquiring, 325 

— related to the stream of life and 

to the shore of materiality, 

323 

— relative, degrees exist for, 164 

— sphere of, 184 

— systematization of, 126 

— ultimate, consciousness as a bar- 

rier to, 207 

— unification of, 256 
Kosmogenesis, the latent geome- 

trism of, 264 

scope of, 237 

Kosmometer, the, 297 
Kosmos, and consciousness, graph 
representing, 271 

— arrangements of the contents of, 

317-8 

— in a state of becoming, 265 

— magnitude of, 308 

— moods of, 311 

— space as the consistence of, 239 

— see Cosmos, 12 



Labor, division of, between the tui- 
tional and the intuitional 
faculties, 193-4 
— mathematical labors, significance 

of, 176 
La Grange, Joseph Louis, 12, 39, 50- 

3, 321, 322 
and the parallel-postu- 
late, 51-2 
Lambert, John, 58 

and the theory of parallels, 

48-9 
Language, the passing of, 357 
Legendre, 70 



Leibnitz' dictum, 26 

Leonard rays, 336 

License, mathematic, permissibility 

of, 38 
Lie, Sophus, 12, 67 

and transformation groups, 30 

Life, analysis incapable of dealing 

with, 319 

— and consciousness as one, 224 
form rooted in pyknosis, 214, 

328 

the fourth dimension, 172 

inaptitude of mathemat- 
ics for, 179 
power to create, 297 

— as agent of creation, 343 

creative agent, 264 

expression, 303 

vassal of materiality, 329-30 

— causative agencies in prolonging, 

123 

— current of, as engendering ele- 

ment, 331 

— deposits of, and the intellect, 173 

— estranged by a radical incongru- 

ity, 325 

— exalting power of, and humanity, 

331,/; 342 

— exhibition of its remains, 282 

— flow of, 265, 297 

— form as an instrument of, 328 

— indescribable signs of, 219 

— infinitive action of, 327 

— intellect has no aptitude for, 231 

— intuitive, values of, x 

— larger life of the Thinker, 196 

— lead of, followed by intuition, 191 

— limits of, fixed by consciousness, 

198,/ 

— most solid facts of, as shadows, 

195 

— motility of, admits of endless 

variations, 35 



382 



INDEX 



Life, movement of chaos in, 307 

— passage of, 201 

— passage through spatiality, 288 

— power to manipulate, 303 

— recurrent movement of, 307 

— totality of egoic, 243 

— undulations in the current of, 329 

— uniqueness of, 41 

— wake of, and consciousness, 205, 

288 

Life-cycle, evolutionary results of 
the, 259 

Life-stream, the, 278 

Light, polarized, and the fourth di- 
mension, 156 

— consciousness as a spreading, 

168 
Light-years and space, 278 
Limits, the sphere of, consciousness 

as, 163 
Line, as generating element, 144 
■ — the straight, a curved, 76 
Lineage of every principle runs 

back to monopyknosis, 213 
Lines, perpendicularity of, in four 

space, 130 
Lobachevski, 55, 60, 66, 87 
Lodge, Sir Oliver, 340-1; 344 
Logic as architect, 118 

— conventional forms of, 262 

— data of, 286 

— intellect follows in the grooves 

laid out by, 294 

— miracle power of, over facts, 288-9 

— rules of, allowability under, 163 
the game of, 164 

— as symbolism of facts, 287 
Logos and the limits of space, 193 

— being of, 218 

— body of being of, 214 

— consciousness of, 204 

— creative, 12, 213 

— the, 216-7 



Lorenz, 83 
Luther, Martin, 181 

M 

Magnitudes, geometry, a study of, 
203 

— non-sensuous, of hyperspace, 72 
Makrokosmic consciousness, 270 
Malebranche, N., 222; see note 
Manifestation and non-manifesta- 
tion, 206 

Manifold, finite, though unbounded, 
70 

— manufactured by the analyst, 77 

— the, 61-2, 172 
Manifoldness, analytical manifold- 

nesses, as mental excitants, 
89-90 

— as a conventional construction, 

262 
an intuition, 318 

— of space, as a near-truth, 317 
Manning, 82, 157 

— and maneuvers in the fourth di- 

mension, 152 
Manvantara, 213, 228-9, 2 3 2 

— as evolution and involution com- 

bined, 13 
Mask, logic as a, 288-9 
Mass-termini, of lines, 236 
Mastery of life over matter, 330 

— of the sensible world, 196-7 
Materiality as a deposit of life, 295 
consubstantial with spatiality, 

33i,/ 

— becoming spatialized, 261 

— characteristics of, 260 

— engendered by kosmic mind, 261 

— engenderment of, 294 

— interpenetrative with spirituality, 

236 

— interstices of, 264 

— neglect of, 320 



INDEX 



383 



Materiality, shore of, 323 

— transmuted into spirituality, 328 
Mathematicians and the definition, 

37 
limitations of consciousness, 

178 
phenomenal world, 1 18-9 

— as prophets, 336 

— respected by humanity, 337 

— the gods of mathesis, 84 
Mathematics, a determinant of the 

quality of the intellect, vii 

— and the criterion of truth, 323 
kosmic intent of the intel- 
lect, 326 

— as symbology, 184 

— Euclidean, 140 

— fails when it encounters life, 337 

— its inaptitude for life, 1 79 
kingdom yielded to kinetics, 

358 

— possesses no judicative power over 

life, 180 

— the orthodoxy of, 34 
Mathesis, and conceptional space, 

203 

— conduct of the intellect in the 

field of, vii 

— definition of, 13 

— domain of, as origin of fourth 

dimension, 130 

— gods of, 84 

— marvelous domain of, 118 

— realm of, not submissive to laws 

of sensible space, 34 

— things of, as emblems of kosmic 

forces, 239 

— world of, 67, 127 

Matter and mind, syncretization of, 
299 

— appulses of life against, 328 

— as a deposit of life, 330 

— honeycombs of, 339 



Matter metamorphosed by life, 
328 

— mind as wedded to, 249 

— movement of, in four-space, 157 

— qualities of, and faculties, 332 

— seven planes of, 212 

— spiritualization of, as end of evo- 

lution, 327 

— totality of, acted upon by totality 

of life, 328 

— unity of, with space, 260 

— universum of, 261 

Matutinal ceremonials of creation, 

221 
Measurability, as property of space, 

62-3 
Measurement of hyperspaces, the 

science of, 73 
Measurements, space determined by 

the number of, 61 

— systems of space, 91 
Measurer of space, w-dimensionality 

as a, 296-7 
Mecanique Analytique, 50-1, 321 
Mechanics of matter, the passage 

from, to the dynamics of life, 

326 

— to biogenetics, the passage from 

357 

— the turning from, to biogenetics, 

322,/ 

Mechanism, the doctrine of, due to 
analytics, 40 

Memories, stored in the omnipsyche, 
258 

Men as gods, 289 

Mental evolution, determinative 
period of, 31 

elaborative period of, 31-32 

formative period of, 31 

Mentality, inner mysteries of crea- 
tive, 91 

— infantile out-feelings of the, 336 



3§4 



INDEX 



Mentality, the principle of, 210, f 
Mentalities, the adaptation of phe- 
nomena to, 184 
Mentograph, the, 121 

— the basis of intellectual conscious- 

ness, 13 
Mesozoic man and his environ- 
ments, 347-8 
Metageometrical investigations and 

the new sense-organs, 354 
Metageometricians and hyperspace, 
238 

proof of rotation about a 

plane, 149-151 

the curved space, 316 

fourth dimension, 145 

key to the mysteries of 

nature, 131 

— baffled by the direction of hyper- 

space, 146 

— dilemma of, 97-8 

— eschew sense-data, 73-4 

— Keyser on the attitude of, 71 

— perceptual obliquity of, 102 
seed thought of, 182 

— turned to idealized constructions, 

263 
Metageometry, as a stepping stone, 
238 

— denned, 13 

— Riemann, the father of, 63 

— the fabric of, 66 
Metamorphosis of matter by life, 

328 

— of the monopyknon, 214 
Metamorphotic stage of space- 
genesis, 228 

Metaphysician, the preserves of, 
usurped by mathematicians, 
141 

Meta-self as medium of kosmic con- 
sciousness, 13, 217 

Methods of the ego, 344 



Methods, scientific, and the ample 
explication of phenomena, 154 
Mikrokosmic consciousness, 270 
Mind, amenability of, to the evolu- 
tionary movement, 28-32 

— and freedom from conception, 28 
problems of physical 

existence, 32 
the new freedom, 33 

— as vehicle of life, 285-6 
wedded to matter, 249 

— coevalism of, and space, 240 

— consubstantial with space, 239-40 

— divine, of the kosmos, 176 

— evolution of, and conceivability, 

24 

— of the duodim, 128 

— powers of, not attained simul- 

taneously, 334 

— profound influence of hyperspace 

on the, 91 

— space as progenitor of, 224 

— three great epochs of the history 

of, 30-31 

— tuitional, the limitations of, 192-3 

— the unity of, with space, 224 
Mind-principle, the quintopyknon 

as the basis of, 221, Jf 
Mirrors, three, and the hypercube, 

136-8 
Mobility, life as, 264 
Molecule, the four-dimensional, 

Newcomb on, 159 
Monadic phase of evolution, 210-1 
Moneron and man, the gulf between, 

28 
Monopyknon, the, 213, 216 
Monopyknosis, defined, 17; 208-10, 

218 
Monstrosities^ mathetic, 202 
Moods of the kosmos, 311 
Morphogeny, 13, 233-4 
Morphons, the creation of, 219 



INDEX 



385 



Motility, the scope of, 131 
Movement, forward, in knowledge, 
336 

— hyperspace as an evolutionary, 

44-68 
influence of Kant on, 49-50 

— of life in chaos, 307 
Multi-dimensional quality of space, 

justification for, 262 
Multiplication of hypotheses, 118 
Mutuality, intuition as promoter of, 

3™, J? 
Mysteries, inner, of creative men- 
tality, 91 

— kosmic, low-lying plains of, 239 

— of life and mind, 326 

the sciences and the four-space 

operator, 131 
Mystery, kosmic, the revelation of, 
foreshadowed in manifold, 77 

— of reaction, 167-8 

space decreases as copes of con- 
sciousness increases, 273,^ 

N 

w-Coordinates, 203 
^Dimensionality, as a phase of 
geometry, 300 

property of intellectuality, 269 

quality of conceptual space, 

14 

— irreconcilable with perceptual 

space, 156 

— no justification for, 235 

— not surprising that the intellect 

fell upon, 296, jf 

— predicates concerning, 158 
n- Space, imperceptibility of, 94 
Nasir-Eddin, 46, 53 

Natural symbols, ideas as, 126 
Nature, representative, of things, 
119 

— the vacuum-stillness of, 339 



Naught to unity, consciousness regis- 
ters from, 165 
Near-truth, 300, jj 

— as variation from criterion of 

truth, 324 

— based upon partial knowledge, 14 

— space-curvature as a, 306 

— space-manifold as a, 318 
Nebulosity, 267 

Necessity, as a bulwark of geometry, 

77-8 
Nerve-body, the, 349 
Neurogram, 14 
Neurograph, the Thinker's scrutiny 

of, 190 
Neurographical communications, 

244 
Newcomb, Simon, 14-5; 66 
and the four-dimensional mole- 
cule, 159 

on the fourth dimension, 125 

Non-Euclideans and the meaning of 

dimension, 97 
things as they actually are, 71 

— dilemma of, 86 

— meaning of the term, 69 

— results obtained by, 88 
Non-manifestation, as antithesis of 

manifestation, 106 

— and manifestation, 206 
Non-methodic advancement of 

human consciousness, 299 
Non-spatiality, cut off from spa- 

tiality, 266 
Norm, as a value assigned by the 

intellect, 167 

— consciousness as a, 15; 161-202 

— of life, found by synthesis, 300 

the natural geometry, 263 

Note, the dominant, of communal- 

ity, 301 
Notions of space, triple presenta- 
tions to the consciousness, 72 



3 86 



INDEX 



Number, as a phase of kosmogen- 

esis, 309 
— numbers as infinitesimals of unity, 

41 
Numericity of being, 309 



Objects and ideas, 126-7 

— passage of, into the fourth di- 

mension, 130 

— of study, identification of con- 

sciousness with, 189 
Obliquity, perceptual, of metageo- 

metricians, 102 
Obscurantism, mathetic, 154 
Occultist and the scientist, 350 
Omnipsyche, 265 

— as agency of kosmic conscious- 

ness, 258 
neglected factor of evolution, 

259 

— denned, 15 

One-space represented by a line, 134 
Ontogeny recapitulates phylogeny, 

27, 334 
Order, kosmic, the difference be- 
tween, and hyperspace, 143 

— mathematical, discovered by the 

intellect, 261 

— the fiat of, 208 

— the totality of, 265 
Orderliness, 266 

Organ, determined by function to be 

performed, 336 
Organs broken up by life into special 

parts, 343 

— the evolution of, 188 
Orthodoxy of mathematics, 34 
Outcome of new adjustments, 256 
Out-feelings of the Thinker, infan- 
tile, 336 

Outlook, the spiritualization of our 
mental, 40 



Papyrus, a hieratic, 44 
Parallel-postulate and the surface of 
a sphere, 70 

— as basis of non-Euclidean geom- 

etry, 54 

stated by Euclid, 55 

Lobachevski, 55 

Manning, 82 

— failures to demonstrate, 48, 82-3 

— in the Elements of Euclid, 16 

— Saccheri's proof of, 47 
Parallels, convergence of, 80 

— infinity of, through a given point, 

70 

— meet at infinity, 88 

— theory of, 47-9 

Passage from mechanics to bio- 
genetics, 357 

— from mechanics to dynamics, 326 

— from three-space to four-space, 

292-3 

— of mathetic contrivances, 241 

— of space, 16, 297 

Pathway of life, sinuosities in, 330 
Patterns of the intellect, ready-made, 

166 
Pentagrammaton of space, 237 
Perceptibility in the Thinker, the 

faculty of, 188 
Perception, domain of, and the sen- 
sible world, 19 

— inner organs of, 352 

— replaced by conception, 255 
Percepts, and the ego, 244 
Period, chaomorphogenic, 208, Jf 

— determinative, 61-5 

— kathekotic, 210 
Perisophical nature of analytics, 

320,/ 
Perisophism, a, 300 
Permissibility of mathetic license, 

unlimited, 38 



INDEX 



387 



Perpendicularity, 132, 177 

— of lines in the four-space, 130 
Phantasy and reality, line of de- 

markation between, 173 

— the world of, 146 
Phantom-ideal, the, 277 

Phase of the world age, evolutionary, 
209 

Phenomena, efforts to explain, on 
the basis of the fourth dimen- 
sion, 129 

— fashioned by the intellect, 199 

— fragmentary interpretation of, 302 

— mind's method of apprehending, 

186 

— physical phenomena amply ex- 

plained by scientific methods, 

154 
Phenomenal, the inverse of realism, 

278 
Philosophy and the criterion of 

truth, 323-4 

— regeneration of, 358 

— systems of, and zones of affinity, 

124 

Phosphorescence, dim, added to the 
unilluminated pool of sense- 
consciousness by the intuition, 
26 

Phylogeny and ontogeny, 334 

— represented in ontogeny, 27 
Physical as embodiment of all pos- 
sibilities, 227 

— phenomena as space-activities, 

229 
Physicality, the principle of, 210, f 
Physics of the brain, 292 
Pi proportion, significance of, 31 1-2 
Pickering on space-curvature, 279 
Pineal gland, energized by the pi- 
tuitary body, 348-9 

the, 344,/ 

not a vestigal organ, 354 



Pituitarial awakening, general re- 
sults of, 355-8 
Pituitary body and the Thinker, 351 

as an organ, 344-35 2 

C. De M. Sajous on, 345 

hypertrophy of, 347 

location of, 246 

the, 344,/ 

Plane, as generating element, 144 

— movement of, into the fourth di- 

mension, 147 

— realism appears to be divided into, 

164 

— rotation about a, 130, 146-151 

— the seven planes of matter, 212 
Planes, a cube as a succession of, 

148-9 
Plasm, kosmic, strivings of, 215 

— psychic, 260 

— the differentiation of, 214 
Plato, 322 

— and the divine geometry, 193, 237 
shadow consciousness, 281 

— on God as geometrizer, 181 
ideas, 34 

Play, sensuous, of the intellect, 295 
Plenum, space as a, 230 

— the universe as a, 107 
Poincaire, 90 

Point, an infinity of parallels through 
a given, 70 

— as generating element, 144 

— position of, 132 

— succession of points, a line as, 

148 

Polarity between concrete and ab- 
stract, 294-5 

Ponderability, as property of hyper- 
space, 63 

Popularization of the fourth dimen- 
sion, 128 

— of the non-Euclidean geometry, 

66-7 



INDEX 



Possibilities in the world of hyper- 
space, 129 

— of four-space, the marvelous, 130 
Postulate-systems, the multiplica- 
tion of, 85 

Powers of mind, not attained at one 

and the same time, 334 
Pralaya, 213 

— as kosmic quietude, 17 
Predicates, all mathematical, not 

justified by the phenomenal, 

128 
Principle, ensouling, 214-5 
Printing press, the invention of, and 

the laws of psychogenesis. 30 
Problem, the fourth dimension as a 

transcendental. 140 
Procedure, the involved, of arriving 

at hyperspace notions, 73 
Process, evolution a continuous, 327 

— space as a dynamic, 307 
Proclus, 83 

Profundities, kosmic, and hyper- 
space, 156 

Progression eastward terminates at 
the west. 306 

Proof of rotation about a plane, 146- 

151 

Propagation of intuitographs, dif- 
ficulty of, 315 

Propositions, geometric, subjunctive 
quality of, 38 

Pseudosphere and the shape of space, 
70 

— as basis of Beltrami's calcula- 

tions, 17 

— the nature of, 64-5 
Psychics of consciousness. 292 

— transmuted, 260 
Psychogenesis, alchemy of, 225 

— kosmic, 272 

— laws of, and the printing press, 

30, 116 



Psychogenesis. mind's place in, 123 

— outgrown phase of, 302 

— outstanding facts of, 131 
Psychogeny, defined, 17 
Ptolemy. 45, 52. 83 

Publicist, the mathematical, and the 
duodim, 128 

— the non-Euclidean, 146 
Pyknon as a kosmic principle, 212 

basis of space-genesis, 1 7 

Pyknosis. 212 

— and involution, 210, 231 

— as a metamorphotic process, 214 

— the seven processes of, 17, 294 



Quality, dimension as commensur- 
able, 140 

Quantities, algebraic, cannot rep- 
resent space, 125-7 

Quantity, consciousness as a tri- 
space, 172 

Quarto d im, 18, 134 

Quart odimensionality, 292 

Quartopyknosis, 17, 208-10 

Quart opyknotic, principle as basis 
of spirituality, 228 

Quintopyknon, 220 

Quint opyknosis and the a priori, 223 

— meaning of, 17; 208-10 

Quintopyknotic principle as basis of 
mentality, 228 



Race, humanity to evolve a new, 358 
— resplendent, 338 
Rajah-Tamas, 220 
Rajah Yoga. 357 

Rational, valueless when unsanc- 
tioned by the intuition, 187 
Reaction, as a mystery, 167-8 
Real, Thinker as a part of the, 201 



INDEX 



389 



Realism and its degrees, 164-5; 2 79 
the Thinker, 315 

— as life, 286 

— consciousness as touchstone of, 

168 

— found in the direction inverse to 

the phenomenal world, 278 

— homogeneity of, 165 

— illusionary nature of, 36 

— infinite gamut of, 36 

— its degrees and the states of con- 

sciousness, 197 

— new scope of, bewilders the mind, 

1 70-1 

— not of mathematical import, 290 

— of concepts and the sensible 

world, 40 
the domain of mathesis, 33-4 

— psychological quality of, 285 

— the thread of, 169 

— three dimensional scope of, 172 
Realities, abstractionizing, 144, 294 

— incomprehensible to the intellect, 

126 

— innermost, of things, 356 

— lesser give way to greater, 195 

— natural symbolism of, 128 

— our, non-existent to beings on 

spirit-levels, 35 

— shaped upon conventional models, 

294 

— supersensuous, 180 

Reality, as life and consciousness, 

173 

— barrier to the cognition of, 189, Jf, 

273 

— comprehension of, 165, 303 

— current of, 322 

— defined as life, 18 

— flow of, 319 

— flowing stream of, 124 

— myriad ways of presenting itself, 

287 



Reality, naked contact with, 193 

— not an inscrutable quantity, 194 

— obscured, 302 

— of the four-space glibly asserted, 

154 

— thread of, 200 

— universum of, 127 
Realms, supersensuous, 338 
Re-becoming, life and consciousness 

as a, 257 
Reciprocity of consciousness and 
realism, 197-8 

— of the manifest and the unmani- 

fest, 217 

Recoil of images upon us, 167-8 

Reflexive development of the intel- 
lect, 254 

Relativity of all knowledge, 101 

Religion and realism, 170 

— the changes in, 358 

Remains of life exhibited to the in- 
tellect, 283 

Remaking of moods by the intellect, 
201 

Replica of consciousness as basis of 
judgments, 205 

Representation, sensuous, compared 
with a shadow, 1 19-120 

Research, improvement in the meth- 
ods of, 125 

Reservoir, psychic, of evolution, 260 

Residuum, the unexplained, 157-8 

Revelational impressions, 29 

Riddle of spatiality and the intui- 
tion, 325 

Riemann, G. F. B., 18, 19, 54, 66-7, 
87 

— and a limited space, 108-9 

the curvature of space, 5 

determinative period, 65 

finite space, 103 

four-space, 27 

manifold, 317 



39© 



INDEX 



Riemann, G. F. B., as inventor of 
the manifold, 86 

— on the bases of geometry, 61-4 
Roentgen rays, 336 

Rotation about a line, 147-8 

— about a plane, 146-151 
illustrated, 148-50 

— intra-corpuscular, 152 
Rudimentary organs, vague func- 
tions lying back of, 343 

Rule in the evolution of faculties, 333 



Saccheri, Girolamo, 47-8, 52, 83 
Sajous, C. De M., on the pituitary 

body, 345 
Schematism, the suitability of the 

present, 320-1 
Schweikart, Ferdinand Karl, 19, 66, 

69 

— on the non-Euclidean geometry, 

58-60 
Science Absolute of Space, 60 
Science and reality, 170-1 

— no longer empirical, 358 

— skeptical attitude of, 349 
Scientist, method of the, opposed 

to the analyst, 84-5 
Scopographic impressions, 19, 190 
Screen, the impregnated, of creation, 

232 
Sea, choppy, and egos, 256-7 
Search, the path of, for spatial un- 
derstanding, 278 
Seb, the god, 215 
Sects, coplanar, 79 
Self and the immensity of space, 223 

— of the universe, 217 

— the kosmic, 278 
Self-consciousness, 270 

— and space, 223 

— as I-making faculty, 243 

— determination of, 302 



Self-consistency, 173 

Sella turcica, the, 345 

Semi-Euclidean geometry, the, 70 

Sensationalists on the intellect, 26 

Sense-data, spurned by the meta- 
geometrician, 73 

Sense-delivery, one, justified by the 
other, 76 
symbolism of, 203-4 

Sense-knowledge, the sphere of, ex- 
tended, 339-40 

Senses, the answer of, to new needs, 
332 

Sensible world, as domain of per- 
ception, 19 

Sensographic impressions, as per- 
ceptions, 19 

Senso-mechanisms of the intellect, 
165 

Sentience, the principle of, 210, f 

Septopyknon, the, 227 

Septopyknosis, as kosmic material- 
ity, 18, 208-10 

Septopyknotic, the, principle, 228 

Seven planes of matter, 212 

Sextopyknon, the, 209 

Sextopyknosis, denned as kosmic 
sensibility, 18, 208-10 

Sextopyknotic, as emotional prin- 
ciple, 228 

Shadow likened to a sense presenta- 
tion, 119 

Shadowgraphs, 281 

Silences, nature's great, 339 

Similitude of agent and principle, 
243-6 

Sinuosities in the pathway of life, 330 

Socrates, 322 

Solitariness of intellectual testi- 
mony, 76 

Space and edict of disorder, 230 

four-dimensional entities, 128- 

130 



INDEX 



39i 



Space and quartopyknons, 219-20 

self- consciousness, 223 

the idealized construction, 144 

— as a dynamic process, 216 
all-mother, 229-30 

an assemblage of spheres, 143 

becoming, 195 

boundary of another space, 49- 

50 

consistence of the kosmos, 239 

curved, 164 

dynamic appearance, 228, 307 

eldest born of kathekosis, 

21S 

engendered product of life, 

308,/ 

evolution, 226, f 

finite extension, 62, 106 

generable quantity, 103 

infinite continum, 70 

intuitional concept, 225 

kosmic order, 265, 267 

nether pole of non-manifes- 
tation, 207-8 

path of life, 293-4 

pseudosphere, 70 

sheer roominess, 115 

system of coordinates, 99 

unbounded extension, 61 

— best study of, the consciousness, 

269 

— conceptual, as basis of non-Eucli- 

dean geometry, 72 

fundamentally distinct from 

perceptual, 72-75 

— elliptical, 67 

— engenderment of, 260 

— essential nature of, 212 

— fabric of, the, lends itself t© mea- 

surement, 99-100 

— four-, 8 

— fractionalized space, 296 

— foundations of, 221 



Space, four-dimensional, 41 

— Paul Carus on, 354 

— fourth dimension of, efforts at 

making it thinkable, 125 

— generability of, by lines, 143 

— genesis of, 227, jf 

— geometric, purely formal con- 

struction, 75-6 

— limits of and the Creative Logos, 

193 

— manifoldness of, 77 

— mind and, the cocvalism of, 240 

— mystery of, 278, JJ 

— nature of, answered to by mind, 

198-9 

— nether pole of non-manifestation, 

207-8 

— non-Euclidean, the characteris- 

tics of, 72-3 

— not a pure form, no 

— passage of, 16 

— path of search for, must be 

Thinker- ward, 278 

— perceptual, irreconcilable with the 

fourth dimension, 156 

— possible curvature of, and the 

non-Euclidean geometry, 76-7 

— primeval, and tridimensionality, 

236 

— problem of devising a, 87-8 

— psychological nature of, 305 

— real, confounded with hyper- 

space, 91 

— Riemann on the curvature of, 5 

— the unity of, 724 

with matter, 260 

Space-activities, as chemism and 

physicism, 229-30 
Space-center, the, 306-7 
Space-consciousness, 225 

— as a direct process, 224 

— mergence of, with the individual 

consciousness, 195 



592 



IXDEX 



Space-curvi.-ri is an arbitrary 
construction, 262 

— 1 s near-truth, 306 

— nature of, 279 

— no need for, 237 

— translation of, in terms cf the 

intellect, 315 
5 : :e-forms, the construction of. 74 
Space-genesis, 20, 232 ; 

— alphabet of. 2 3 - 

— completion of, 226 

— norm of. 1 74 

— pyknon as a basis c :". : - 

— symbolism of, 208, Jf, 235 
Space-manifold, the, 279 
Space-measurement, and reality, 

Euclid's system of , 06 

— systems of, 201 

Space-mind and archeological an- 
dences, 193 

kzz—'.tizi. : : - 

the Thinker. : - y : - ; 

— attainment of, 225, 231 

— granaries of. ; • 

— fie realization of, 238 
Syaze-ir: :ess :ie. 5:5 
Space-realities, hyperspace as a 

stepping stone to, 238 
Spark, the omnipsyche as a, 258 
Spatiality and the intellect, 263-4 

— consubstantial with materiality, 

20, 331, 1 

— cut-off from non-spatiality, 266 

— interpenetrative with materiality, 

236 

— nonconformity of, with logic, 164 

— riddle of. and the intuition, 325 
— , rise of, 180 

— Thinkers outlook in, 316 

— world of nascent, 232 
Spencer, Herbert, 322 
Spinoza, 322 

Spiritism, the phenomena of, 154 



SliZr 25 



Spiritualists, the claim of, regarding 
the fourth dimension, 1 

Spirituality, materiality transmuted 
into, 328 

— principle of, 210,^ 

— seeds of. : : : 
Spiritualization of the flesh, 227 

man's mental outlook, 40 

matter, the end of evolution, 

r-- 

Squares, the form of, in the 

Euclidean geometry 
St 1 Eai :e. 7 ye. 66 
Stage of pyknosis, 208,^ 
— , quartopyknotic, 210,^ 
Stanley, Hiram M., on 

i;.~ami£— . ::_-- 
States, the two, of 

one unaware of the other, 192 
Straight, a, determined by two 

-::.:~ ;: 
Stuff, etherealized four-dimensional, 

::c 
Sum. 2-r_li:. -; 

— triangular, less than two right 

:~:'t= - : 
Super: :r_:e:::5. :s in :u::: it:, pus. 155 
Super: izzeprj.^". 25 :ue iz:ui:::ri.. 

Superperception and automatism of 

lie izzztltzz. :z 
Suuerperreuruii. uie 74 
Superuniuu. :ue. :" 
S"eier.: :rr. : >:. zls 
Symbol, psychic, and the brain, 

190 

— words as symbols of ideas, 126-7 
Symbolism of life, 286 

reurcrraphicai 

186- 

realities, 128 

sense-deliveries, 204 

space-genesis, 208, Jf 



INDEX 



393 



Symbology, depicting planes of con- 
sciousness, 270 

— of mathematical knowledge, 
203, I 

Synchronous evolution of faculties 
and properties of matter, 332 

Syncretization of mind and matter, 

299 

Synthesis and the norm of reality, 

300 
Synthetic school of geometry, 56 



Tactographic impressions, 190 

Tartrate and a component in the 
fourth dimension, 156-7 

Task of mental evolution, the su- 
preme, 169 

Taurinus, 56-7, 69 

Telescope, its effect upon conscious- 
ness, 298 

Tendence, intellect as scientific, 
165-6 

Tendencies, destined to flower as 
faculties, 89-91; 332 

Tendency to fragmentate, the intel- 
lect and, 296 

Tesseract as represented by Cams, 
136-8 

— illustrated, 133 

— the, 20, 145 

— the elements of, 135 

— the four coordinates of, 132 
Testimony of the intellect, incap- 
able of comparison, 76 

Tetragrammaton, the, 308 

Tetrahedron, the, 135 

Thalamencephalon, the, 345 

Thales, 44 

Theory of knowledge, 290 

Thingness of objects and the im- 
possibility of comprehending, 
122 



I Things, the dual nature of, 119 

value of, 161 

Things-in-themselves, the identity 

of, 353 
Thinkability of the fourth dimen- 
sion, 125 
Thinker as a pure intelligence, 184, 
248 

engenderer of the intellect, 291 

principal to the ego, 243, f 

six times removed from the sen- 
sible world, 201 
the source of the intuition, 26 

— and concepts, 244 

his approach to the kosmic 

mind, 202; 223 

larger life, 196 

treatment of the ego, 246 

of sense-impressions, 74-5 

realism, 315 

space, 193 

a segment of reality, 171 

— the, and the amplitude of the de- 

grees of realism, 198, 270 

— and the barriers of consciousness, 

207 

brotherhood of man, 301 

degrees of realism, 175 

manipulation of concepts, 

255 

nature of space, 268 

neurograph, 190 

new faculty, 335 

perceptive faculty, 188 

pituitary body, 351 

— denned as the spiritual man, 21 

— his consciousness and the ether, 

280 

dependence upon the ego, 245 

method of contacting the ex- 
ternal world, 1 89-1 9 1 

outlook upon the universe of 

spatiality, 242 



394 



INDEX 



Thinker, his schematism of cogni- 
tive powers, 206 
sphere of awareness, 283 

— the difference between, and the 

intellectuality, 256 

evolutionary needs of, 336 

upreaching yearnings of, 336 

Thinker-ward, space must be sought 

in a direction, 278 
Thinking, abstract, elevating influ- 
ence of, 33 
Thought, prolonged abstract, bene- 
fits of, 249 
Time, as an aspect of consciousness, 
224 

— divested of timeliness, 279 
Todhunter, 46 

Tool, fashioning, life as a, 264 
Topography of the intellect, 312 
Touchstone, consciousness as a, 168 
Tracery of connection between ideas 

and objects, 126 
Trafficking in mental constructions, 

154 
Transcendental, the realm of, and 

mathematicians, 142 
Transfmite as a limit, 122; 216 
Transfmity, 21-2 
Tree-reality, 126 

Triadic phase of evolution, 210-1 
Triangles, angular sum of, 78, 88 
Tridim, the, 22, 129, 134, 147, 177 
Tridimension, the, 147 
Tridimensionality and primeval 

space, 236 

— and the space-mind, 193 

— a quality of perceptual space, 22 

— mastery of the phenomena of, 172 

— the sufficiency of, 198 
Tripyknon, the, 212, 213 
Tripyknosis, 17, 208-10, 219 
Truth compared with facts, 288 

— criterion of, 5, 323-4 



Truth, facets of. 284 

— kinds of, 324 

— Jogic does not illuminate, 287-8 
Tuitive, the, and the intuitive facul- 
ties, 192-3 



Ultima Thule, the, 207 
Unbounded, the, as a finite exten- 
sion, 70 
Undulations, three, in the current of 

life, 329 
Unfoldment, mental, and meta- 

geometrical discoveries, 131 
Uni-circle, the, 308 
Unification of all knowledge, 256 
Uniqueness of real space, 95 
Unitariness of all existences, 241 
Unity, as end of analysis, 42 

— death as a fourth dimensional, 

159 

— kosmic, integers as fractional 

parts of, 41 

— of mind and space, 224, 230 

— the new realization of, 225 
Universality, geometric, based upon 

the formal character of as- 
sumptions, 77-8 
Universe, a glorified, 226, 268 

— as a full, 308 
plenum, 107 

— and the seven planes of matter, 

212 

— character of, fixed by conscious- 

ness, 162 

— limited and conditioned, 127 

— not a pure form, 108 

— the perfected, and the circle, 310 
theory of the mechanical ori- 
gin of, due to analytics, 40 

— unity in the, 301-2 
Universum of appearance, 187 

— of life and consciousness, 257 



INDEX 



395 



Universum of reality, 127 

space, 269 

Unknowable as a symbol, 194 

— the darkness which shuts out 

the, 207 
Unodim, the, 129, 134, 176-7 

consciousness, 163 

— denned, 22 

Upraisement of matter, 329 
Upspringing of a new faculty, 333 



Vacuum-stillness of nature, 339 
Validity of mathematical conclu- 
sions, 101 
Value assigned by the intellect to 
the sensible world, 167 

— assumptional, vitiating influence 

of, 163 

of the ideal, 275 

Vanity of fragmentary evidences, 
204 

— of intellectual method, 325,^ 
segmenting space into many 

dimensions, 299 
Vassal, life as the, of materiality, 

329 
Vehicle of life, form as, 330 

mind as, 286 

Vicissitudes of kosmic evolution, 

21S 
Vision, the inner, 356 



W 

Wachter, 69 

Wallis, John, 46, 83 

Weismann, note, 260 

Words, as symbols of ideas, 126-7, 

204 
World and the child mind, 121 

— as instrument of consciousness, 

199; 298 

— fabric of, and geometrism, 261 

— impossibility of objectifying the 

fourth dimension in the per- 
ceptual, 124 

— of phantasy, conditions of, iden- 

tified with sensible realm, 146 

— phenomenal, and hyperspace en- 

ties, 128-130 

— sensible, as a carpet, 196 

— the domain of a perception, 19 

genesis of, 167 

World-plasm, kathekotic, 267 

— as elemental essence, 329 



Yoga, Rajah, 357 

Youth of the earth, the nebulous, 

335 

Z 

Zollner, and the claims of the spirit- 
ualists, 154 
Zones of Affinity, 22, 124 
Zoometer, the, 297 










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